A German historian coined the term shortly after the fall of Constantinople to the Ottomans as a way to distinguish between classical Rome and the medieval state. The term stuck, especially because the Byzantine period was so looked down upon as a decadent, unworthy remnant of the republic and the empire of Augustus.

It was never a term used by the Byzantines themselves, though they did refer to Constantinople itself as Byzantium and its inhabitants as Byzantines for stylistic reasons (it was fashionable to make classical references in literature).

With the reorganizing of the Roman empire between East and West; the eastern half was centered upon the city, as we know it today, Istanbul. Prior to Roman control though Istanbul was known as its Greek name, Byzantium. From this name the city would be renamed to Constantinople after the emperor Constantine in the 4th century. There was still an association though with the old name of Byzantium though, which would lead to the Byzantine Empire, having its capital at Constantinople. I hope this lent some understanding.