At peak time Roman Empire was superpower of the world but It have two Empires at same time.
My Question is who was more powerful then other. and who have the command.
How we can differentiate them.
I know they both are allies but who had control over other.
As @Mark points out, the Empire was too large to be efficiently managed from a single central point in the 4th century: at that time, the complete area consisted of provinciae, full of Roman citizens who had to be protected (no client state to serve as buffer), and the external borders were under high demographic pressure ("barbarians" on the other side of the Rhine/Danube boundary were more numerous and more organized than in the 2nd century). This called for fast reaction time (militarily speaking) which could not be obtained with pure centralization (or at least so it was thought). Diocletian thus got himself a co-emperor, and both of them had each a co-caesar. However, Diocletian was still the formal master; Maximian was not his equal.
The real split was done in 395 (Theodosius I was the last "common" emperor) and things were already going sour at that point. Honorius evacuated Britain only 15 years later, the same year that Alaric sacked Rome; Adrianople was already in the past. The East and West empires did not have much time to be allies or enemies. We may note, though, that the split occurred on a linguistic/cultural boundary between the Latin and Greek worlds; the same split was much later reenacted in 1054, when Catholic and Orthodox churches went separate. This is a profound division which still has some relevance in modern geopolitics (e.g. remnants of that division were still at work in the Croatia-Serbia war twenty years ago).
In any case, the Eastern half was richer, with a much bigger population, and a lot more trade with the rest of Asia. Notably, when Diocletian instituted the Tetrarchy, he chose the Eastern half as his specific dominion; similarly, Constantine ruled from Constantinople, not Rome. The Western empire was mostly rich in area, which would make in particular future France agriculturally wealthy and a demographic giant throughout Middle Ages, but this was for later times. In the 5th century, most of that area was still covered with forests.
When the Roman Empire collapsed in 476 AD/CE, much of the Italian peninsula-(including Rome), as well as many countries to the North and West of Italy proper, fell into "The Dark Ages". Countries, such as Austria, Southern and Western Germany, parts of Switzerland, much of France and the near entirety of England, as well as the majority of Italy, fell into a cultural and political decline until the beginning of the 2nd millennium AD/CE-(the traditional starting point of the Late Middle Ages). Although the Roman Papacy was still quite powerful throughout the Middle Ages, much of its power and influence was regional and primarily confined to Italy proper, as well as to the historically less sophisticated and economically underdeveloped lands to the North and West of Italy.
It was a different historical reality for the Eastern half of the Roman Empire. Although it was actually the Emperor Diocletian who officially separated the Roman Empire into Western and Eastern zones, it was essentially the Emperor Constantine who literally relocated the Administrative capitals of the Roman Empire to Milan in Northern Italy, as well as to the centuries old Northern Greek city of Byzantium, which he renamed to, "Constantinople" around 306 AD/CE. It was Constantinople, that would follow in Rome's imperial footsteps........ towards the East.
The commercial and geopolitical strength of Constantinople was important, due to its close proximity to Asia, especially the famed Silk Route-(which begins in Constantinople, from a European perspective), as well as having close access to the Mediterranean and Black Sea regions. Although the Byzantine Empire faced invasions from Slavs, Vikings, Goths, Arabs, Crusaders and ultimately, the Ottoman Turks, it was an extremely powerful state which had the capacity to counter such invasions over an 1100 year period-(with the notable exceptions of the Crusaders, as well as the Ottoman Turks).
Comparatively speaking, Byzantine Constantinople absolutely dwarfed Early Medieval Rome in terms of its political power, economic and commercial wealth, cultural refinement, religious influence and historical longevity. Early Medieval Rome, as well as its former Western colonies, were in the distant shadow of Constantinople and the greater Medieval East. Even with the short lived accomplishments of the Carolingian Renaissance ushered in by Charlemagne in Aachen, Germany during the late 700's, the balance of power during the Early Middle Ages.......was in the East and Constantinople, was its Center Point.
