The terms "Western Roman Empire " and "Eastern Roman Empire" are modern classifications. The ancient romans did not consider the empire to be split into two independent states, they thought of it as a single state ran by two imperial courts. The split was purely administrative and was originally created by Diocletian around 285 CE when he instituted the beginnings of the tetrarchy system in an attempt to avoid the total collapse of the empire after the chaos that was the crisis of the third century. The split is first seen through Diocletian's rule of the east and Maximian's rule of the west as equal Augusti.

There are several instances after the demise of the tetrarchy where Rome was ruled by a single Emperor in charge of both the eastern and western empires. Examples include Constantine the Great, his son Constatius II, Julian and Jovian. The co-emperor system was reinstated by Valentinian I when he made his brother Valens emperor of the eastern empire. Theodosius the Great also ruled alongside his brother Valentinian II.

It wouldn't be until near the end of Theodosius the Great's reign that the empire was once again controlled by a single man. After his victory over Eugenius at the Battle of the Frigidus River Theodosius seized control over both the east and the west and became the last emperor to control the entirety of the Roman Empire.

After Theodosius the two emperor system returned and remained until the collapse of the Western Roman Empire. The two empires were more or less on friendly terms, with the Eastern Empire providing military support to the Western Empire during the battle of Cap Bon in 468 CE.

I don't know about the maps someone else will have to answer that part of the question.

TL;DR The Eastern Roman Empire and Western Roman Empire were thought of as two parts of a single Roman Empire and Emperors of East and West were usually seen as equals or at the very least subordinates.

There’s not much to add to u/WispyWalrus ‘ point about the basic conceptual unity of the Late Roman Empire. Its citizens didn’t think of the Western and the Eastern Empire as separate states but as two halves of on unified res publica. There are some ancient sources that use terms like Hesperium Romanae gentis imperium (Western Empire of the Roman people) but they are few and relatively late like Marcellinus Comes who wrote around 520 AD about the “fall” of that entity.

The other part of your question about the maps is quite interesting as it projects modern notions onto the Romans which they didn’t actually share with us. They didn’t really produce maps with clearly delineated borders inscribed on them. Doing that would go against their notion of living in an imperium sine fine, an empire without end. The Roman Emperor was the only legitimate ruler in the world and all others were only governing at his sufferance. The limites that the Romans erected at some of their borders weren’t supposed to delineate the ending points of Roman sovereignty but were only used to regulate the migration of border peoples. The Romans regularly intervened in the internal politics of surrounding societies and thought those interventions to be absolutely legitimate.

Case in point would be the tabula peutingeriana, the only extant Roman map of the world that I know of. It may look a little strange to modern eyes as it clearly isn’t supposed to be a topographically correct map. Instead it is organized around the important roads traversing the empire. I sometimes compare it to a modern subway map which also isn’t supposed to convey an accurate topography but instead a more or less easily understandable web of connections. And it certainly doesn’t feature any borders.