This question probably popped up due to the current concept of plagiarism that we think of it this way. We're quick to claim "this was copied" over similarities, while the traditional development of things was in fact closer to evolution. Concepts don't exist in a vacuum and naturally grow from previous works or ideas.

There are commonalities, but for a different reason than you'd expect: people back then had no concept of secular as we do now. They saw their gods as a part of the world and, while philosophers would discuss their nature, their religion would actually be builit into their society. In fact, we often think of the Athenian myth, as it builds most of our knowledge of Greek beliefs. You should remember that even amongst the Greeks differences between cities could be much, much more pronounced than differences between Christian denominations. The very structure of these religions actually reflected the Greek and Roman political organization.

They took their religion very seriously, and their religious priorities reflected the Roman society: to them, Jupiter reflected the chief institutions of Rome, and so on. You should also be mindful of how the Roman religion was very... fluid, so to speak. It was not uncommon to have noticable differences between observance of religious ritual between families within the same city. Notably, while we think of "ordained" priests and such, religious duties were part of offices we'd see as "secular" today; a father of a family was typically a sort of a religious head for his relatives.

The general thing is, these differences are actually reflected in the wider culture rather than the Greco-Roman hodgepodge that we've come to take for granted. This is because, both the Greeks and Romans, had scholars and they were looking for commonalities. The "connection" we normally think of here is not because there actually was one, but rather because the Romans looked for parallels. And so, they would "map" other gods to theirs, just like the Greeks. In fact, we only take them as "so similar" because we've been raised thinking of them as "vaguely the same thing". If we were not equipped with all the comparative writing, we'd probably take a long time to come to these conclusions. The linking has already been done for us - it's a bit like mathematical proofs; it makes sense when you look at the result, but figuring it out on your own is difficult. If someone shows it to you, it seems obvious. If you have to notice it and prove it, not so much.

This also reveals an interesting point, showing how central they were: specific gods could be taken as general cultural contexts they'd translate. In a way, to them, the difference between Ares and Mars was the difference between War and Krieg. In this case, we could think of the general concept, but even though we'd seemingly be looking at the same thing, the execution would be different (To illustrate: someone on reddit pointed out how the British have mechanized infantry, while the Germans have mechanized infantry. Technically the same name, but fundamentally different approaches.)

This way was actually how the ancient societies understood each other fairly well: they did not get hung over doctrine or minutiae; to them, the gods were a nebulous expression of nature and aspects of life. They would sometimes describe gods that barbarians worshipped not by their actual names, but rather by the Greek or Roman gods they resembled. They would also adopt gods from other pantheons and mold them to fit their systems of beliefs, most notably from Egypt. edit: As /u/Thoth_the_5th_of_Tho pointed out, this did not mean they were peaceful: rather that they applied broader approaches to understanding beliefs. Instead of fighting over minor differences, they looked for and thought in terms of common ground. It's fascinating how the Greeks were also able to map a large chunk of Egyptian beliefs to their own, seeing as they were even more distinct - and that's a whole different topic I now kinda want to look into myself.

The best way to describe it is probably "cultural translation" - the reason we think of these similarities is because the Greeks and Romans themselves went to great lengths to estabilishing them. What's nowadays considered irreconcilable doctrinal differences would qualify as simple regional difference to the ancients. These religions developed in geographic proximity, in similar conditions (i.e. mediterrenean climate). They developed parallel and they started comparing themselves to each other.

Probably of note is the fact that Roman religion was a lot more practical: it focused on the Eternal City and the "now", so to speak. It was intrinsically linked with Rome, while if you consider the Greek myth, you'll find that the Romans didn't particularly concern themselves with a proper creation myth, while the Greeks had theirs. It's also pretty clear how Greek myths are hellenocentric, while Roman myth is remarkably grounded around Rome and its way of life. Furthermore, due to the Greek fragmentation each city had significantly different practices, just look at the popular Sparta vs Athens comparison. We like to put them in the same "mythology bag", but their observance and customs almost couldn't be any more different. Again, we like to underestimate how much their religion came from and influenced their daily lives and customs; it was a feedback loop, so not only were there differences in focus - the "general" Greek mythos was the result of each Greek city assimilating the stories of their neighbours, but each city remained unique in some practices noted by the others (for instance, the infamous Spartan infanticide). As they were polytheistic, they were also not strictly contradictory or mutually exclusive.

For instance, consider the Vestals: they were definitely a Roman feature. The Olympic Games were distinctly Greek (yes, they were a highly religious ceremony!). Jupiter, unlike Zeus, was not married to his own sister. The Romans considered Mars a noble god, one of the most important, while the Greeks were at best ambivalent towards Ares. You could easily point out that Ares reflected the Trojan War, while Mars reflected the Legions. In Rome, priesthood appears to be more of a state post than in case of Greece, largely owed to Rome's empire (as opposed to the polis). We should also remember the evolution of their religion: for instance, over time the Romans developed an Imperial cult on top of the more traditional polytheistic beliefs. In general, the large amount of "similarities" we think of is not exactly that similar. It's the result of synthesis of hundreds of years of religious evolution with ideas flowing freely and local beliefs differing even between households.

Obviously, this only scrapes the surface, as we're conditioned to focus on the similar: the best way to find out more is to look into each equivalent separately, as trying to synthesize just a comparative list for Greek and Roman beliefs and religious customs is actually an endaevour that could easily take multiple volumes.

Source: A recollection from a collection of essays on everyday life from various researchers, specifically on Ancient Romans and Ancient Greeks. A fascinating read and a great, broad overview for anyone interested in the topic; if you can get these in a language you can read, by all means get your hands on them.

L'Uomo Greco, Jean-Pierre Vernant editor, Gius Laterza & Figli Spa, Roma-Bari 1989 (Polish edition, Bertelsmann Media Sp. z o.o. 2000) and its counterpart on the Romans (still not sure where I stuck it)

L'Uomo Romano, Andrea Giardina editor