The distinction comes down to how the state itself was understood by contemporary people. In Rome, the early emperors depicted themselves as officials of the Roman Republic invested with unique power and preeminence. The title of princeps claimed by Augustus, for instance, merely signified that he held auctoritas principis (civil authority over and above that wielded by the other citizens of Rome, including the heads of government), within the structure of the existing Republic. So there was no formal declaration or even acknowledgment that the name, or more importantly the constitution, of the state itself had changed. In practice, it would have been obvious to politically-engaged Romans that the rules of the game had changed significantly, especially as power passed within the imperial family, but the transfer to non-family members (e.g. the Flavians and then the Nerva-Antonine line) wouldn't have signified an interruption of the state any more than the initial assertion of "preeminent citizenship" would have by Augustus.

In imperial China, conversely, the conception of the state was inherently tied to the ruling family. Indeed, the name of the state, as understood by contemporaries, would actually have been "Great Tang," "Great Song," "Great Ming," etc. The names were chosen by the founding emperor of each dynasty, and traditionally corresponded to a fief or command held by the founder prior to their ascent (e.g. the Tang founder had been the Duke of Tang and the Song founder had commanded an army with the Song title). The state apparatus was all centered on the person of the Emperor as the ruler of "all under heaven" (tianxia), and when the imperial line changed, so did the construction of the state (even if, in practice, a lot of the institutions remained consistent, e.g. the aristocratic bureaucracy from Sui to Tang). All in all, a different notion of an emperor than in Rome, where he was an official superimposed above an existing hierarchy.