Roman mythology is ... complicated. Influences from Greece was presented on the Italian peninsula prior to the foundation of Rome. So it isn't surprising that the Romans adapted Greek myths into their own religion. But Greek myths were not the only ones. Just north of Rome settled Etruscan tribes, who also had their own religion. Over time, Rome conquered more and more land from the Etruscans (and later all over Italy, Europe and Northern Africa), and myths and deities from the conquered tribes were added to the own mythology. So, the Roman Mythology is one part original creation by the Romans, and one part amalgamation of of other mythologies around them.

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For the second part, we can find many examples. The Roman Jupiter is basically the same as the Greek Zeus, Juno is Hera, Mars is Ares, Diana is Artemis, and so on. The Romans took one of their already existing gods, then tried to add as many attributes as possible from the corresponding god. Original Roman deities are rare, but they exist.

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One of them, for example, is Janus. For the Romans, he was a two-faced god, god of the beginning and the end, god of transitions. He was often named in the beginning of prayers, so he appears to have been very important in the Roman world. And this importance persistent, even after the Greek gods were adapted. As god of doorways, frames and gates he was also a part of the numina, a group of spirits inhabiting different areas within a household. And the cult of the househld was very important in Roman rituals.

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The reason why there is no Greek counterpart is simple. Both myths that tell the origin of Janus state that he actually lived, either as a ruler alongside the early Roman king Camesus or around the time of the mythological founding of Rome by Romulus. This also makes him one the earliest gods the Romans had, and according to some, creator of the universe and ruler of the gods.