While we refer to them nowadays as the Eastern Roman Empire and the Western Roman Empire, the Romans themselves did not make such a political distinction. It remained one empire, but was governed by two emperors with separate jurisdictions. Of course, people are people, so the emperors themselves had their own power struggles, not unlike siblings (which they often were, e.g. Arcadius and Honorius in 395). But nominally they ruled the same empire.

Diocletian initiated the first such division in 285, after decades of civil unrest, military action, and fights for the throne. Succeeding emperors would reunite and redivide the empire several more times before the final split in 395, after the death of Theodosius I. The Western empire then struggled through 80 years of shaky alliances with northern tribes before Odoacer took control of Italy.

After the fall of the West, the Eastern Empire often tried to win back Roman lands- note that this was viewed as a reconquest, even provinces in Italy and North Africa had never been a part of the Eastern Empire. They considered themselves Romans even until the fall of Constantinople to the Ottomans, 1000 years after Rome was captured. For instance, check out these coins with the slogan "Deus Adiuta Romanis," "God Help the Romans."