I already made my comments in the other thread, but it was the legal successor state.
To go into more details, there were several Latin speaking Roman empires. The first was the old Republic, was was destroyed not really when Caesar marched on Rome as the history books have it, but when Sulla marched on Rome earlier and implemented a new constitution. The Sullan republic broke down within decades, and Octavian did a restoration job (constitutionally his changes were actually less radical than Sulla's) that is known as the Principate. The Principate then broke down in the 260s and 270s, after which a completely new government was put in by the series of Emperors between Aurelian and Constantine, with most of the changes associated with Diocletian.
The post-Diocletian government is called the Tetrarchy or sometimes the Later Roman Empire, though there is no good name for it. It was a new system of provinces, military commands, and bureaucratic posts, topped by two central governments, one based in Northern Italy (first Milan and later Ravenna) for the western provinces, and one based near the Bosphorus (Nicomedia and then Constantinople) for the eastern provinces. The Senate, the magistrates, and some of the traditions of the past made it through into the new system, and Latin remained the official language. The borders were also close initially to the Principate borders. However, this empire was neither based in Rome nor were the people running it Roman nor Italian.
Justianian's government and provinces were the same as Diocletian. The eastern part of the Later Roman Empire lasted until the seventh century, with no substantial changes other than the obvious one from the switch from the earlier modified paganism (itself a break from the paganism from earlier periods) to Christianity. The western part fell apart piece by piece in the fifth century, with no obvious point that you can point to where it ended (the traditional 476 date is as good as any other alternative), with the eastern government later taking over parts of the west.
In the seventh century, the eastern empire replaced Latin with Greek as its language, reorganized the bureaucracy, and imposed a system of fused civil-military commands, including the themes, on the old provincial system. Territories outside of a heartland in Asia Minor were lost throughout the century, and cities dwindled away, even Constantinople losing a good part of its population. As with the west in the fifth century, its impossible to point to a specific date where these changes took place. Any time between the deaths of Heraclius and of Justinian II is a good choice. Unlike in the fifth century, the office of the Basileus or Emperor itself was never abolished, so there is more continuity.
The Later Roman Empire is the legal successor state to the Early Roman Empire, and the Byzantine Empire is the legal successor state to the Later Roman Empire, or at least its eastern half. The Frankish kingdom actually has a decent claim to be the heir to the western half.