It wasn't - the Byzantine empire is the eastern half of the Roman empire, and, as such, existed (in various states) until 1453. The debate as to whether we should call Byzantium Rome has been raging since Gibbon. In its own time, Rome existed certainly through the end of the Constantinople-based empire. They called themselves Rhomaioi and the emperors were Roman emperors; other polities and tribes did business with the Romans and not the Byzantines. Still they were different from the ancient western empire. East Rome spoke exclusively Greek after around the 500s and the empire was very much more theocratic - the emperor was daily involved in the rituals at the Hagia Sophia, and the Patriarch (leader of the church) was second in importance in the city. But they were always Romans - even when the Ottomans took over, the Sultan Mehmet II took a title over the new lands, calling it Rum.

Sources: the writings of Constantine VII clearly show that they were always Romans in their own minds, especially the De Administrando Imperio

Dates can be rather arbitrary. Traditionally, the end of the Western Roman Empire is dated to 476 due to the abdication of Romulus Augustulus. However, Julius Nepos, who was emperor 474-475, was reinstated from 477(?) to 480.

[Emperor of the East Zeno] requested the senators to take Nepos back - and Odoacer, too, was bidden to do the same. Odoacer, in fact, predictably took no action to reinstate him in Italy, but the resumption of gold coinage in Nepos' name at Italian mints indicates that Odoacer (who controlled him) at least ostensibly accorded him the recognition that the eastern emperor had requested. The tradition, therefore, that the western Empire ended with Romulus Augustulus needs at least formal modification...

Michael Grant, Roman Emperors, p.326

Later Western historians tended to downplay the East, giving it the name of Byzantine and thinking that the true end of Rome was in 476. Exactly why is speculation. Gibbon was somewhat derogatory to the east, as /u/greatbrokenpromise points out. Maybe the Great Schism and subsequent troubles in the Christian church(s) played a part. Many in the West had looked down on the Greeks, even when they were part of one Empire, like Cicero's 'Greekling' jibe in On Brutus shows.

This has changed somewhat in recent years as realisation that the East had much to offer and had a legitimate claim on the Roman throne. I think John Julius Norwich's excellent work on Byzantium has something to do with the East's rehabilitation.