Often in school I was told that the Roman's got all their good ideas from the greeks. Obviously this isn't true (THE MANIPLE SYSTEM). However I heard that Tacitus said the Germanic tribes worshiped Mercury. Which suggest a link between the Romans and Germanic tribes. One thing that the germanic tribes, roman civilisation and greek cities share is the Indo-Eurasian language group. Did the Romans inherit their mythology from the proto-indo-eurasians, separate from the Greeks.
The Romans inherited an Indo-European pantheon, which manifested with considerable changes throughout the distribution of the various Indo-European languages. When Romans encountered pantheons that seemed familiar, they attempted to fit "those people's gods" into their Roman counterparts, taking them to be local manifestations of their own supernatural beings.
Because the Greeks had such a rich literary tradition and they had gods that were easily recognizable as counterparts of the Roman tradition, the Roman's embraced that literature, but they did not "just copy the Greek mythology." They had their own traditions, which they freely augmented with literature that seemed to enhance their own beliefs.
At times, the Roman attempt to fit foreign pantheons into their own failed. Belief systems mutate over time, and the amount of separation between the Germanic world, for example, and the Romans was considerable. Thus, the Romans believed that they recognized Mercury in the northern worship of Odin, but that was certainly a false fit since the Roman deity Mercury and the Germanic deity Odin (with all the various spellings for that entity) probably did not have a common origin.