What is the distinction between the Byzantine Empire and the Roman Empire?
The Byzantine Empire is distinguished from the Roman Empire largely by geography, culture, and religion.
The Byzantine Empire had its origins in the Roman Empire. Beginning in the 300s AD, Constantinople essentially became the capital of the Roman Empire in the East. This split with Rome became official in 395. At that point, it was still the Eastern Roman Empire, not the Byzantine Empire.
The Byzantine Empire came to differ from the Roman or Western Roman Empire in a number of ways. Geographically, it was centered in the East, with the capital in Constantinople which is in what is now Turkey. The Byzantines did briefly conquer and control Italy, but for the most part, this was an eastern empire. Culturally, the Byzantine Empire was Greek, not Roman, with Latin eventually falling out of use completely. In terms of religion, it was built as a Christian empire from its beginnings, instead of being an empire built on pagan foundations as the West was.
All of these are ways in which the two empires are distinct even though the one gave rise to the other.