For as much as people say, the Byzantine Empire is not the Roman Empire.
The Roman Empire ended when West and East were divided into two empires. After that division, we had two Roman Empires (one was more Greek than Roman, but nevermind).
I read that they created both Empires to be "easily administrated". If that were the case, the Eastern Roman Emperor was under the rule of the "only and true" Roman Emperor. But he wasn't!
Both Emperors considered the other as an equal, a brother, showing that they were on the same level. And the laws as acts that were passed, for example, in the WRE, where not valid in the ERE, and vice-versa.
So they where two distinct Empires that shared a common foundation, but they where extremely different from one another. One was Greek, the other Roman, one had the army organized into A, B, and C, the other into D, E and F, etc, etc.
Regarding the topic, the "Roman" culture was lost after the fall of the WRE. When I mean culture, I do not mean art, music, language, etc, but the feeling of being a Roman. The WRE was divided into multiple kingdoms and A LOT of "new people" from the lands beyond the Rhine. People living in Northern Italy did not considered themselves as Roman, not did the Southern Italy people. So the feeling of "Oh boy, I wish I could unite all the Roman people of Italy into one big Republic as did my great-great-great-great-great grandfather" was lost.
But as an achievement, uniting all the former Roman provinces into one nation is a nice one...