Oh boy, this topic is my specialty, so sit tight.

A lot of the confusion about the difference between the Greek and Roman gods has to do with how the Romans viewed religion. To put it succinctly, the Romans focused on the actual practice of religion and appeasing gods, rather than on who the actual gods were. They perceived it as in certain locations and/or with certain practices, someone could call upon the power of some deity. This is why some Roman prayers began with “To the god or goddess located here…” Honestly, the Romans didn’t care much about mythology beyond how it was related to Rome. Dionysus of Halicarnassus has a famous quote:

“Let no one imagine, however, that I am not sensible that some of the Greek myths are useful to mankind, part of them explaining, as they do, the works of Nature by allegories, others being designed as a consolation for human misfortunes, some freeing the mind of its agitations and terrors and clearing away unsound opinions, and others invented for some other useful purpose. But, though I am as well acquainted as anyone with these matters, nevertheless my attitude toward the myths is one of caution, and I am more inclined to accept the theology of the Romans, when I consider that the advantages from the Greek myths are slight and cannot be of profit to many, but only to those who have examined the end for which they are designed; and this philosophic attitude is shared by few. The great multitude, unacquainted with philosophy, are prone to take these stories about the gods in the worse sense and to fall into one of two errors: they either despise the gods as buffeted by many misfortunes, or else refrain from none of the most shameful and lawless deeds when they see them attributed to the gods.” II.20

What this quote is saying is that the Romans will accept all manner of practices and rituals and myths, but they don’t particularly care about where they came from. The Romans are more concerned with the deity being happy (to not get smited) than anything else.

I’m going to look at the two example deities now, one of which I wrote a paper on last year and one whom I should be writing a paper on instead of this comment.

The goddess Tanit, of Carthage, underwent a really interesting transformation when the city of Carthage was eventually rebuilt into a Roman colony—she began to be worshipped under the name Juno Caelestis. Note that the Romans attached epithets to a lot of the “classic Roman deities” to associate them with a specific aspect: such as Juno of Veii (from the Italian city of Veii), Jupiter Dolichenus (from Syria) and Mars Ultor. Originally, Tanit was the consort of the major deity of Carthage, Baal Hammon, but not worshipped very much on her own. As the Romans strongly disliked the practices of Baal Hammon (look up the tophet if you’re interested), the main god of Carthage became Juno Caelestis. Interestingly, despite the usual marriage being Juno/Jupiter, Baal Hammon was often called Saturn, because his practices were associated more with Kronos/Saturn eating his children. In this way, we can see that calling the foreign goddess Juno doesn’t actually fit that well with the other myths. Rome is just calling the goddess whatever they find convenient. Furthermore, the worship of Juno Caelestis didn’t change much since Carthage’s destruction. People in Carthage still worshipped on stele in a very similar way to before, just with minor word choice changes.

The other example goddess I have is the Magna Mater. The Magna Mater was a mother goddess who originated in Phrygia, in Asia Minor, before moving eastward to Greece and finally Rome. In 205 BCE, Rome officially imported the goddess in a large ceremony, after the Sibylline books said it would help protect Rome during the Second Punic war. The titles of Magna Mater are interesting: the main one is Cybele, but she was also known as the Idean Mother, Dindymusian Mother, and Berecynthian Mother. Finally, she was associated with Rhea, Oikomene (the world) and possibly Gaea. Ida, Dindymos, Berecyntia, and Cybele are all mountains in Phrygia where she was originally worshipped. She is only called Oikomene in one place, and I personally feel it is a misnomer. The real question is, why was she called Rhea in some places. The answer has to do with how the Romans placed her in their histories. The phrase Magna Mater literally means “Great Mother,” and not a lot of mythology existed for her except as a mountain goddess. The Romans generally interpreted this title to mean that she was the mother of the gods, for whom the closest comparison was Rhea. However, the Magna Mater played a role in the legend of the founding of Rome (protecting Aeneas because he was also from Ida), which wouldn’t fit with the mythology of Rhea. The Romans didn’t care. They made associations that helped them understand how the goddess might react and want to be honored.

Overall, the only thing the Romans cared about was the favor of the deity. There’s two good ways to understand how the Romans viewed gods in general. The first is that the gods are a bunch of balloons high up in the sky with strings going to the ground. The string tells you that there is a god present, but it doesn’t tell you which god or what they’re going to do. You don’t want to get smited, so you have to treat the god with respect and courtesy. Furthermore, if you pull on the string in a certain way, you can ask the deity for favors and communicate with them. The second way is that the Romans understood talking to the gods like contracts. The gods would do something for you, but only if you made certain sacrifices and libations.

The other thing I’m not mentioning is the origins of Rome. Rome believed that they were originally from Troy (that’s the whole descendants of Aeneas and the Aenied thing), but they also knew about the Etruscan gods and were influenced by them. This means that their understanding of the Greek gods was slightly changed.

To sum up the answer to your question, it’s not that the Greeks and Romans had the same gods—it’s that the Romans took the Greek gods and added it to their extremely large pantheon.

The FAQ has a link to "Were the Roman gods literally transplanted from the Ancient Greek tradition or did Rome coincidentally have a pantheon of Gods that lined up with the Greeks and they made the comparison later on?" , with a top-level answer by /u/idontplayoboe but also noted by /u/cleopatra_philopater .

But much more could be added; this is not to discourage discussion. Further questions, data, and debate are welcome.

Both Greek and Roman pantheons stem from a common ancestor that lived somewhere in the Eurasian steppe around 3000 BC. This common ancestor is referred to as the proto-indo-european culture. Around that time they started expanding and bringing their culture everywhere around them

This is according to a theory first popularized by Marija Gimbutas and greatly expanded upon in modern scholarship. The theory is these were some of the first individuals to domesticate the horse on a large scale. They then used the increased trade and technological benefits to expand their culture into neighboring regions. For a long time. All the way to Ireland one way and all the way to India the other. Plus eventually most of Europe and even to the borders of China. At the very least this movement is done through language, culture, and religious tradition if not outright population displacement/intermixing. Whether this movement was done by migration, trade, warfare or a combo of the three will forever be debated.

So not only do the Romans and Greeks have a common ancestor that leads to the similar religious deities, this is why Celtic, Germanic, Slavic, ancient Iranian Zoroastrianism, a lot of the Hindu tradition, etc have "similar" gods and stories. Of course the gods are not exactly the same even if they have a common ancestor because they often develop apart from each other for thousands of years over thousands of miles. And often they develop new gods or adopt gods from people they contact which are not found in other "similar" pantheons.

But a very simple way to show the evolution is through the language. Proto Europeans had a hypothesized sky god called Dyḗus Patḗr (sky father) in their (theorized) language. Over thousands of years of migration from the eurasian steppe, the Greeks had basically changed how they said it. The Greeks dropped the second half and kept Dyḗus but changed the D to a Z for Zeus. The Romans did the same and changed it over their evolution and migration from the eurasian steppe. The Romans dropped the D of Dyḗus for yḗu Patḗr or Iupiter (remember theirs no J in Latin so Romans would have said it closer to YḗuPatḗr)

Very basic version of the evolution but hopefully that will get you interested in exploring more of the Indo European scholarships that's around. Its a huge field of study and conversation that pulls from almost all modern aspects of history, archaeology, linguistics, genetics and comparative mythology. Its cutting edge work on one of the largest movements in human history that had almost an incalculable affect on human history and culture.

A good book on this topic is The Horse, the Wheel, and Language: How Bronze-Age Riders from the Eurasian Steppes Shaped the Modern World by David W. Anthony