So as a kid I was taught that the greek and roman gods were the same just with different names, but I find that premise difficult to accept now- for a couple of reasons:
The hellenic & Italic branches of the indo-Europeans were separated long enough for the Romans to have developed their own indigenous mythology, sure the borrowing of gods happens- but then which gods from the greeks were direct borrows, which were indigenous, and which were borrowed from other areas?
And if I’m missing valuable context here, and the romans did directly borrow greek gods- is it because it’s an artificial construct of both having parallel gods that were similar enough to each other from the ancestors of both peoples that later authors failed to recognize the distinctions, but instead just viewed it all as greek because they had no knowledge of earlier ancestors ( such as the indo-Europeans?) As the premise of a direct borrow doesn’t make particular sense, gods are borrowed from elites, and greeks were the junior partners to Rome during the Roman era- since they were obviously conquered- so why would an elite borrow gods so heavily from conquered peoples?
Or are their parallel mythologies just an o oversimplification that might have missed out on centuries of syncretic faiths heading in to rome from all across the empire, of which greek was just a large source?
With no deep knowledge on the topic, but till a more complete answer...
There have been noted obviously relations and connections between the gods of Greeks and Romans. One of the first Romans who wrote on such topics was Cicero [1st c. BCE] in his De Natura Deorum [On the Nature of the Gods]. This work is full of ancient greek references [like Homer & Hesiod with commentaries] and connections. The gods seem to be considered as common. Maybe something to be expected to some point as the ancient greek literature was richer at the time; however, it shouldn't be ignored that Cicero had inter alia a greek education, that maybe had an effect.
If the names can tell something:
Cicero [in Cic. N.D. 2.67] considered for example the goddess Vesta's name as greek coming from the equivalent greek Hestia [=Εστία]. More modern approach has indicated this relation, too [eg. Etymological Dictionary Of Latin by Michiel Vaan, 2008, p. 671]. However there were goddesses like Ceres and Venus whose the greek equivalents had different names; Demeter and Aphrodite respectively.
Maybe Ceres is a good example.
She was the goddess of agriculture. Cicero thought that the name was derived from latin gero that stands for holding [the corn]. I've read also about a possible etymology from creo->create. And finally that comes from a word-root for feeding, nourishing [eg. Etymological Dictionary Of Latin by Michiel Vaan, 2008, p. 109]. This last meaning could be considered close somehow to the greek one, as Demeter stands for mother-earth. However it's a different word.
But further, in greek mythology the greek goddess Demeter had a daughter, Persephone. In Roman mythology she seems to be adopted and transformed into Proserpina, as Ceres' daughter again [Cicero, Ovid]. It's typical that Ovid [Met. V, v. 425ff] uses both names for the same deity. Cicero considers it a greek loan [in Cic. N.D. 2.66]; and seems to be an accepted approach nowadays too.
This couple Ceres-Proserpina made me think of a possibility that there were independent to a degree similarities in their mythology on an archetypical base, without being able to know exactly why. But later, after the communication of their civilizations, they were noted and led to possible loans and relations.