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First, I would like to clarify that Diocletian was emperor prior to Constantine, and his Control Center was located in Nicomedia.
I would say the break between Rome and Byzantium, historically, occurred between the Death of Justinian and the reign of Heraclius. This marked the end of several trully Roman facets in Byzantium. First, the offical language of government went from Latin to Greek, second the emperor's court began to radically change from its Roman procedures and positions to uniquely Byzantine positions, third, the empire stop depending on foreign mercenaries and created a new system of raising an army (the Theme system), and finally, the Emperors on the Bosporus gave up any hopes of reclaiming the Western Empire.
From the respect of the Byzantines, they always were the Roman empire, though thier language changed, the fact was even at Rome's height the East primarily spoke Greek, so it wasnt too much of a change for the people. Second of all the people claiming lineage back to Rome, ie the Holy Roman Empire, only the Byzantines could state a legitimate claim based on sucession back to Augustus Caesar. In addition, the Byzantines called themselves the Romanoi, or Greek for Romans, to the Byzantines, the Empire didnt fall it just lost some dead weight.
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