The Romans and the Greeks shared a common inheritance from the Indo-European pantheon, and because of this shared origin of many of their gods and associated stories/attributes, there were many similarities. Religions tend to change over time, so there was some drift before the Roman world fully embraced the Greeks, but the similarities were obvious to the Romans.

The Romans tended to try to see local reflections of their gods in the people they conquered, and this was more or less easy when dealing with other people with a shared Indo-European inheritance, but in many cases, drift made these identifications strained. It was much easier with the Greeks because the drift was less pronounced, and since the Greeks had a highly-regarded body of literature dealing with their own gods, it was easy for the Romans to adopt that literature, which in turn influenced their own literature and beliefs/stories.

So the bottom line is that the gods were as you say "original to the Romans and they just happened to be similar to what the Greeks believed," but the Romans were also opened to allowing the eloquently-told Greek stories to influence their own believes since they perceived the Greek pantheon as manifestations of the same gods.