also, was there a reason why Apollo, seemingly alone, appears in both pantheons?
In brief - for educational terms, it's a good-enough simplification for school purposes. But yes, there were important cultural distinctions between the two.
I'm away from my books at the moment, so I'll do this one 'off the cuff' and lay down some suggestions for further reading: if you want to ask follow-up questions, I'll be able to get to those in a couple of days with the bibliography in front of me.
One thing that's often glossed over in school courses - and I say that as someone who teaches them - is that 'Greek Religion' and 'Roman Religion' (or even, perish the thought, 'Greco-Roman Religion' are not simple, single, fixed ideas. Both varied considerably over time, where you were, and who you were. I'm going to talk a bit about Greek religion first to lay some groundwork, and then try to describe what happens when it collides with Roman religion.
At this stage, we need to introduce the idea that the 'same' god could look quite different in different contexts. It was normal (particularly in the Greek world) to address a god not only by their name, but also by an 'epithet' - a short descriptor that gave a bit of detail about the god, but also reflected the particular aspect or characteristics of them that you were calling on. Apollo (we'll get to him properly later) is a great example - at Delphi, you would probably address him as Apollo Pythios ('Apollo of the Pythia - his oracular priestess at Delphi) and primarily conceptualise him as a god of prophecy who might advise and help you. On the other hand, if your city was struck by a plague, you might sacrifice to him as Apollo Ekbolos ('Apollo who Strikes from Afar') and conceptualise him as a terrifyingly inscrutable force for vengeance. On one level, they're the same god, but they also feel rather different.
You might also have learned in school that the 'big gods' - the Olympians and so on - are only the tip of the religious iceberg: both Greek and Roman religion are full of 'small gods' of rivers and mountains, semi-divine heroes and other extremely local figures that mean that religious observation can change considerably from region to region, or even village to village. Even when talking about the greater gods, different people worship them to different degrees - most obviously, the people of Athens treated Athena as one of the most important of the gods, while she would be a relatively minor religious force in most other Greek cities.
This means that the 'international', 'official' Greek pantheon (which was never an idea that anyone in the Greek world, ever, would have accepted) was only ever a kind of lingua franca - people around the Greek world could understand it and use it as a common point of reference, but it didn't really reflect anyone's actual, day-to-day experience of religion. Straight away, we're dealing with a religious landscape that is incredibly fragmented, and where any attempt to generalise about it smooths down the reality.
EDIT: So is the Roman pantheon - on which more detail here.
Roman religion and Greek religion have a common source - they both descend from Indo-European belief and folklore, which included key concepts like the association between the gods and the sky, the ultimate descent of the gods from the union of the sky-god and the earth-goddess, and a set of 'divine twins' who were the sons of the sky-god and act as heroes and rescuers. However, Italic (and so Roman) religion developed distinctive aspects with no parallels in the Greek world. These include the belief in the lares and penates, small, unnamed gods of hearth and home which protected both people's homes and the state as a whole. They also included the practice of augury (reading the future from the flight of birds), and various beliefs about the spirits of the dead, such as the benign manes and the restless, malignant lemures. Though they've been somewhat neglected in the scholarship until recently, these played a huge part in the day-to-day religious life of the Roman world.
The Romans engaged in a practice called interpretatio Romana, where they 'interpreted' others' gods as reflections or misunderstandings of their own, and so worshipped them either as alternate names or as epithets of their own gods. Hence we see dedications made at ancient Celtic shrines of Sulis to 'Sulis Minerva', as the Romans 'interpreted' Sulis as their goddess Minerva. It's an interesting question to think about what was going through the heads of people worshipping at that site - are they just worshipping their old gods, and paying lip service to the Roman names? Or is there something fundamentally significant about doing even that in a Roman temple, in Latin, surrounded by Roman statues? Interpretatio is certainly an act of imperialism - by 'claiming' gods and situating them within the Roman pantheon, it turns their worship into an act of integration into Roman culture, and reduces any opportunity to use those gods as a locus of non-Roman identity, or as a focus of resistance to Rome.
It's really in the Late Republic and the Augustan period that we see a huge boom of Roman texts talking about the gods, and most of these have a vested ideological and/or literary interest in playing up the similarities between the two. In most cases, they come from writers trying to establish a place for Rome within the literary and cultural world created by the Greek classics, particularly the works of Homer, and to situate Rome's foundation and identity within the heritage of Greek mythology. Virgil's Aeneid is the great example - I've written another post here about how that poem engaged with a complicated oral tradition, and how Virgil invariably made choices that emphasised the glory of Rome's foundation but also which constantly worked to create links between the Greek legends of the Trojan War and the Italic myths of Rome's foundation, even when that didn't make the most logical or narrative sense.
While there is always more to say about a topic, u/RainyResident had an excellent answer here: How did the Greek and Roman gods become so similar?
Roman Religion is complex, and it changed over time. It's not a pure 1:1 parallel. In the Early Republic, Rome's gods were a mix of their native cults and a number of imported Hellenistic figures, though not necessarily from the "official" Greek Pantheon.
T. P. Wiseman notes in various of his books from "Roman Drama and Roman History" to "The Foundation of Rome: Myth and History" that ritual seemed to have been more important to early Romans than actual beliefs. He notes as an example The ceremony of Lupercalia. This was a story of Luperci sacrificing a goat. and the Romans staged an entire ceremony around this act. It wasn't so much faith in a god so much as faith in the story. He writes:
On 15 February, two days after the Ides, there took place at Rome the mysterious ritual called Lupercalia, which began when the Luperci sacrificed a goat at the Lupercal. There was evidently a close conceptual and etymological connection between the name of the festival, the title of the celebrants, and the name of the sacred place: as our best-informed literary source on Roman religion, M. Terentius Varro, succinctly put it, ‘the Luperci [are so called] because at the Lupercalia they sacrifice at the Lupercal … the Lupercalia are so called because [that is when] the Luperci sacrifice at the Lupercal’.
What is missing in that elegantly circular definition is the name of the divinity to whom the sacrifice was made. Even the sex of the goat is unclear — Ovid and Plutarch refer to a she-goat, other sources make it male — which might perhaps imply a similar ambiguity in the gender of the recipient. Varro does indeed refer to a goddess Luperca, whom he identifies with the she-wolf of the foundation legend; he explains the name as lupa pepercit, ‘the she-wolf spared them’ (referring to the infant twins), so I think we can take this as an elaboration on the myth, and not much help for the ritual.
This is something you will often find contrasted between Rome and Greece. Greece depicts statues with Gods that have names. Romans depict scenes who's God could be supplanted with any local cult. For Romans, it was more the ritual, the act, the experience, than whatever God was there.
We see this even further elaborated in how we remember Rome's early Religion: Through ritual story.
In many ways these stories, and the rituals attached to them, are Early Rome's Pantheon.
On the topic of Numa up there, Rome's later monarchs quickly began to view that these older stories were "Numa's religion", as Rome's 6th king called them. The pre-Hellenistic deities part of the founding mythos. It's interesting really. Rome acknowledged they had abandoned their founding deities for Greek ones. Servius Tullius is credited as having done this shift for Greek mythology, having built a Temple to Fortuna and Diana, and essentially culled the elder-gods to the confines of history. There is perhaps here an echo of the Greek Titans, viewed as elder godsd. Randall Hansen and many others make note of this phenomenon in classical religion He writes of the Titans as the former gods:
TITANS (GREEK TITANES) Older family of gods who preceded the Olympians. As a group the Titans are the older gods, the former gods, in contrast to the Olympians, who are the younger and present gods.
Was this an attempt by Rome to mirror the Greeks? IDK. Not relevant.
In this era, we see Rome start to prop up the Greek mythology as their own. Mars suddenly becomes the figurehead of the military. King Numa being a man who contested with Mars in favor of Athena, to de-militarize Rome. There's an intertwining of the two mtyhos to create a codified Roman origin story. Later on, at the height of the Republic, this shifts to adopt a number of Greek gods to center the stories around. Here's the closest you will find to a 1:1 ration, but even here, you will notice a distinct difference. Whereas Greece's myths focus on the gods, Roman myths focus on the men through whom the gods work. For example, Rome's founding myth involving Aeneas focuses much more on the man than Apollo, the God who favors him. And you can kinda tell this is a later adoption because he's somehow supposed to be the founder of Rome, but Romulus is too, at least as a free city state. There's a shift to make Romulus the descendant of Aeneas in order to make it all make sense, but this is my opinion.
Thus you see that 1:1 argument is maybe not so true and only really a few fleeting moments in Roman History posses it.
Towards the Imperial era, you will again see a shift in religion. The Romans start propping up the idea of "Sol Invictus", whereby all these greek gods are merely incarnations of a central god. This itself likely a product of the popularization of Stoicism and the concept of the Divine Spark. You also see a shift in Roman religion to view the Greek gods as mere men rather then deities. Former kings. This likely was pushed by Jewish and Christian writers who were attempting to de-myth the Greek pantheon, but it was also a growing belief among Romans and Greeks as well, started by Euhemerus who in the 4th Century BC began writing that all gods are dead men of long gone past. Pagans such as Diodorus Siculus and Christians such as Eusebius of Caesarea write in support of this view. So there seems to have been broad consensus that this was the case. Eusebius writes:
That those are no gods whom the common people worship, is known from this: they were formerly kings, who on account of their royal memory subsequently began to be adored by their people even in death. Thence temples were founded to them; thence images were sculptured to retain the countenances of the deceased by the likeness; and men sacrificed victims, and celebrated festal days, by way of giving them honour. Thence to posterity those rites became sacred, which at first had been adopted as a consolation. -De idolorum vanitate
And Diodorus writing:
According to the myth which the priests give, the gods had their origin in Crete, and were led by Zeus to Panchaea at the time when he sojourned among men and was king of the inhabited earth. In proof of this they cite their language, pointing out that most of the things they have about them still retain their Cretan names; and they add that the kinship which they have with the Cretans and the kindly regard they feel toward them are traditions they received from their ancestors, since this report is ever handed down from one generation to another. And it has been their practice, in corroboration of these claims, to point to inscriptions which, they said, were made by Zeus during the time he still sojourned among men and founded the temple. -The Library of History, Book V.
In that same era you see the propping up of other new gods. The city of Rome begins to be deified as a god herself. Tacitus writes in the Analls that a temple to the city as a god was built in some places:
They had also been the first, they said, to build a temple in honour of Rome, during the consulship of Marcus Porcius Cato, when Rome's power indeed was great -Tac. Ann. 4.56
And of course there is the growth of Mithridates, Christ, and others throughout the Empire.
Thus you see that fleeting moment of a 1:1 ratio just as quickly flees in favor of whatever was popular.
All this is to say, at the height of Pagan Rome, it appears the Romans viewed the Olympians as either chief gods, or for the Stoic intelligentsia, great kings of the past touched by some divine spark they called Sol Invictus. However these were supplanted deities placed in the background of an already established religion. The Romans definitely viewed the Greek Gods as their masters and ancestors, but they were the backdrop of their founding mythological heroes who had a special set of honor. Men who found favor of these masters, but the truth is those masters could be supplanted for whatever pantheon was popular, and sometimes often was. These elder heroes, known as the gods of Numa are shown as subordinate to these Olympians, and yet capable of challenging them and perhaps even beating them the same way the Olympians beat the Titans. But this is a general outline and throughout their history there were shifts.
Just remember, for the Roman, ritual played a much more important role than faith.
EDIT:spelling
Yes and no. As you suggested, this is presented for simplicity. While it's true that the main gods can be related to Greek equivalents, I'd like to think it's more complex than just the Romans stole the Greek gods and re-named them. And the Roman's are well known for essentially stealing pieces of other cultures and then branding it as their own, so it's not totally wrong, however they adopted from other cultures as well.
For instance, if we look into the history and etymology of the Roman god Minerva, thought to be the equivalent of the Greek's Athena, she actually has origins from the Etruscan culture. Etruscan culture also existed in Italy (Tuscan region) prior to when the Roman's were first thought to have settled (and perhaps originated from if we don't take Virgil's Aeneid at face value). Here she was known locally as Menrva. It is thought that over the centuries she took on more and more of the common epithets & ritualistic traits known from Athena, eventually becoming the Minerva/Athena that was born from the head of Jupiter/Zeus. It is also thought that perhaps when Rome was founded - likely by a few Etruscan men in the area - Menrva came with them to form the backbone of the newly forming Roman culture. Her worship can also be dated back to the founding of Rome in the 8th-7th century BCE where she formed part of the Capitoline Triad along with Jupiter and Juno (who also likely came from the Etruscan pantheon where they were known as Tinia and Uni, respectively). Who she represented then is largely unknown as we don't have much written history pertaining to the early days of Rome, but it's likely that the early Minerva of Rome's foundation was completely different from the Minerva of the Republican era, and this Minerva was different to the Minerva of the Imperial era. It's likely that throughout the centuries, the gods were adapted and altered as the Romans came into contact with other cultural groups, and even more likely that the gods were Hellenized by the Romans to give them a more courageous & interesting backstory like the Greeks boasted so that the Romans could look even more prestigious and favored by the gods than their Greek rivals.
Rome was known to let it's newly conquered people retain their culture and local pantheon of gods as long as they recognized Rome as their head of state. So it's assumed that as Rome came into contact with these cultures, the 'cool gods' of these conquered states were of interest to Roman citizens and would gain traction and eventually become known as 'that Roman god' instead of 'that Greek god' or 'that Egyptian god' where it originated. Sometimes this meant that a god was literally adopted from them (such as Greek's Apollo or Egypt's Isis) or it could mean that one of Rome's existing gods took on more of their traits and epic mythology (like Minerva, Jupiter, and Juno). However, not all Roman gods morphed into Hellenized gods over time - the Roman gods Quirinus and Janus, who were very popular during the Republican era of Rome, seem to be distinctly Roman and don't directly have a Greek or other counterpart. Their history also dates back to the early days of Rome suggesting they may have been invented and then popularized as gods by it's early kings.
We should also remember that Rome thrived for over 1000 years as a culture. Over this time Rome had the opportunity to come into contact with a number of cultures and adopt them conquer them to grow their empire. The Romans also admittedly forgot their roots over the centuries as written history wasn't common like it is today and most of history was reported orally throughout the centuries.It's thought that as the stories were passed down, they become increasingly exaggerated and the truth slowly deteriorates from the narrative. It's much easier and more fun to tell the tale an epic adventure of a god than it is say that a god was adopted from that local tribe 200 years ago. This is why Virgil's Aeneid was so popular since he presented a narrative about Rome's foundation that was otherwise unexplained to the Roman's of the 1st century, and he presented a Rome that was founded by a man from Troy, directed to this land by a Greek deity, and told in the epic format invented by the Greeks.
First up, a preface that I've just re-read part of Beard, North and Price's superb Roman Religions to try and avoid any howlers on this.
When you're taught the "Roman pantheon" and the "Greek pantheon" in school, in some kind of general introduction to "Classical civilisation", you're really only being taught about maybe twelve or so gods where there are parallels in terms of overall places in their pantheons. So a useful example from archaeology that Beard/North/Price bring up is Vulcan, where a Roman religious site from the 6th century BC features Athenian pottery decorated with imagery of Hephaestus, which suggests that there was some element of identification between the two even going that far back.
However, although there is some level of identification between the "Olympian" gods (as they're often referred to, although that's very much a Hellenocentric term!), it falls apart when you look at Roman religion as a whole. Take a relatively significant god such as Janus; certainly the poet Ovid considered him to be a "uniquely Roman" god, although that is overall debatable given potential influence from "near Eastern" cultures. He has no Greek equivalent, certainly.
(incidentally Ovid is worth bringing up in this context because a lot of the confusion around the Greek and Roman gods stems from what is often also taught alongside their names and roles - myths relating to them. Many people know these myths because of Ovid's retelling of them in the Metamorphoses, for example, where he takes Greek myths and uses the Roman names of the gods in telling them - but they are Greek, not Roman.)
Anyway, leaving aside the vastly different overall make-up of the Greek and Roman pantheons (not to mention things like the cult of Mithras!), the spirit of your question is essentially around the Olympians. I can't and won't go into detail on all of these, but let's take Minerva as an example, usually identified with the Greek goddess Athena. Before I do so, I need to introduce a term you might not be familiar with (apologies if this is patronising!)
The Olympians in the Greek and Roman worlds have their big, overall responsibilities that you've probably covered in school - "Dionysus is the god of wine" or whatever. They were often worshipped in different aspects, however, where a particular form/responsibility of the god was honoured. A useful example to think of with this in the Greek world that springs to mind is the sanctuary of the Acropolis and its dedicated to the goddess Athena. Within (or technically just outside...) that sanctuary, you have temples or statues dedicated to Athena as "Athena Nike" (goddess of victory), "Athena Promachos" (goddess of battles), "Athena Parthenos" ("the maiden" goddess) etc. etc. These little addenda to their names are known as epithets, that show the responsibility being worshipped.
In my copy of Adkins and Adkins's Dictionary of Roman Religion, the basic listing of Minerva is that she was the goddess of crafts and trade guilds, before adopting part of the martial aspect of Athena Promachos - so relatively similar. You'd also note from that listing the three different festivals that worshipped her (none of which correspond to, say, the Athenian Panathenaia). She is listed under three further epithets - Minerva Capta, Minerva Flanatica, and Minerva Medica. The first two of these refer to specific Roman events - the capture of an icon of the goddess in Falerii, and her apparent patronage of Flanona in Dalmatia. I don't believe medicine is generally associated with Athena but don't know for certain.
So using the above, we can already see that an Olympian god can have multiple Roman-specific aspects rather than just being completely cloned from the Greek version (also worth noting, of course, that Minerva's name is essentially Etruscan in origin...)
I could go on further, particularly in regards to things like Apollo, but in the middle of writing this I came across /u/XenophonTheAthenian's excellent response in a thread from five years ago, which explains things better than I probably would go on to do (or already have done!). Short answer to the Apollo question regardless - it's 'cos of the Etruscans adopting Apollo from Greek religion, also Apollo isn't really his Greek name anyway, it's Apollon.