I know it was mentioned before, but the term "Byzantine" did not come into widespread use before 1800-1900. The rulers of both empires considered themselves to be "Roman," and if you want to go further, the Latin Empire, which was founded by Crusaders, was also considered to be "Roman" because the leaders felt that they were justified in bringing back the glory of the Roman Empire and linking that with Christianity.
And Ivan the Great of Russia even married a noblewoman from the Byzantine Empire, and claimed that he was the new "Roman" Emperor.
Also, if you look at South America, the inhabitants call themselves "Latin" American because Spain had been under Roman rule, and they had become culturally assimilated.
So I'm sorry, but to tell you the truth, there is no real such thing as a "Roman," "Byzantine," or "Latin" Empire.