Ares and Mars are not identical, Poseidon and Neptune are not identical.

In general Ares was looked down upon by the Greeks, not readily worshiped (few native Greek temples existed to Ares), and seen as a god of a subject they would rather avoid. On the other hand Mars was one of the favorite gods of the Roman state religion, held in higher esteem than any other gods other than Jupiter and Minerva and their culture saw war as a glorious endevour. They built numerous temples to Mars and placed them in prominent places.

The reason most people believe they are identical is because the polytheistic religions readily conflated gods accepting stories from other cultures where the characters were similar. The Romans in particular readily accepted Greek mythology identifying their native gods with those of the Greeks, but they added the Greek stories to their own native identities, they did not replace them. The Romans continued to worship their gods in a more organized state religion way not found in the Greek world. We do not get this from the myths. Its like asking what is the difference between Catholicism and Baptists but only looking at the fact they both use the Bible. You need to look at how the gods were actually worshiped, what structures were built to them and their status in society.

The two cultures were able to easily conflate their gods because in general they had similar sources. Both of their pantheons grew out of the older Proto-Indoeuropean pantheon until the cultures separated in prehistory. The Greeks were initially influenced by the preexisting non-Hellenic peoples of the peninsula, bringing in or adding to gods like Poseidon, Dionysos and Apollo from those people or eastern ones.

The Romans added very little until contact with the Etruscans and then Greeks well after those cultures had established great civilizations, writing an literature. Where the Romans lacked gods they added from those older prexisinting civilizations, where they didn't they retained their customs. Then the Romans went out into the world and imprinted their ideas on both those civilizations so there becomes this circle of where does the influence start?

You are commenting on what is known in the ancient world as interpretatio graeca (or romana). That is the people used the ancient practices of the Greeks (or romans in the latter) to understand the mythologies and religions of other peoples. So are Ares and Mars identical, no- not at all. Ares has very little cult in Greece, and was seen as a god of war, of rage. While Mars was a god of just war and of agriculture. Every year in October the Romans sacrificed a horse to Mars partially in commemoration of the harvest, but the harvest and war were inexorably tied since the Romans had seasonal war time and would go to war and then return to tend to their farms.

Moving outside of Rome/Greece you can see it in other cultures as well. Sulis is a British god that was equated with Minerva, and you can see that in the shrines in the thermae in Bath where there are devotions to Sulis Minerva. Or how Odin in the Norse pantheon was equated to Hermes. Or how many romans thought that the Jews worshiped the god Zeus Sabazius (Sabazios being a horse riding god from Thrace). It is simply a way for the Romans or Greeks or the Natives to understand the mythos of other locales.