The right relationship with the gods
All ancient religions were aimed at having the right relationship with the gods, not believing in the right things. The Greeks and Romans were very similar in this way. There weren't really any heretical beliefs around this time because religion was often a more personal matter (although state religion certainly existed). Each had their own way of relating to the gods. They may have followed some general guidelines, but there were no competing religious sects. It is the first major attribute that the Romans borrowed from the Greeks.
The gods: different and identical
It is true that the Romans borrowed many gods from the Greeks, but they almost always gave these gods their own local or national spin. An example is the capitoline triad of the gods: Jupiter, Minerva and Juno. These gods are similar to Zeus, Athena and Hera. These gods were regarded by the Romans as gods of the Capitoline Hill and gods of the city of Rome in particular. Jupiter was not the same as the worshiped Zeus in Greece.
The right way to worship
The Greeks and Romans sacrificed to the gods in the same way. They made offerings of animals, other food, or wine libations. The sacrifice coincides with the pagan belief of having a good relationship with the gods. These offerings were meant to appease anger and to offer thanks for trying to influence the unknowable will of the gods.
The Odyssey and the Aeneid
The similarities between Greek and Roman myths are emphasized by the similarities between Homer's Odyssey and Virgil's Aeneid. Virgil was a Roman poet who wrote his epic poem during the reign of Emperor Augustus. The writer of the Odyssey, Homer probably wrote around 850 BC; several centuries ago. The two works are very similar: they both occur after the Trojan War and they both follow the journey of a former soldier of the war. The layout of the two poems is similar, but once again the Romans took a Greek tradition and made it their own. The Aeneid speaks of the discovery of the future site of Rome by Aeneas and his interactions with gods and men along the way. While Homer and Virgil present stories born out of the Greek myth of the Trojan War, Virgil uses this classic setting to tell the story of the founding of Rome.
In conclusion, the Romans borrowed a lot from the Greeks, but we cannot ignore how they took the Greek myths and religion and made them their own.