Do you consider the Byzantine Empire and the Roman Empire to be the same?

@tagsorwhatever: While the Byzantines spawned from the Eastern Roman Empire, the fact that they lasted so long after the fall of Rome means they deserve to be considered their own thing. That being said, the Byzantines were one of the inheritors of Rome's Imperial power and for better or for worse kept things Roman in their corner of the world which has lasting impacts on Europe and the Middle East even to this day.

They considered themselves Romans, but from our point of history it does refer to two different empires

No, completely different.

Depends on what aspect is being talked about