After the administration was split (a few different ways were tried, settling on eastern/western halfs), the western half declined, failed to stop incursions from the Germanic tribes and eventually Rome was sacked a couple times.
The eastern half thrived, and even reconquered Rome and Spain, under Justinian. This was a continuation of the exact same government, no fall from power, just a loss of some territory. They lost Rome back fairly quickly and were controlling various parts of the Eastern Mediterranean, Eastern Europe, the Middle East and North Africa on and off until they entered terminal decline after 12-1300 or so, and Constantinople finally fell to the Turks, with the last Emperor dying in the battle in 1452.
Whats somewhat sad about the western half of the empire is that the final 100-150 years were just riddled with things that were entirely avoidable and other stupidity like the treatment of the Gauls.
But they had a much harder time dealing with the huge influx of displaced Germanic tribes due to the (Mongols iirc?). The western half of the empire started down the road to things like castles and other behavior like large estates that directly fed into the middle ages. Diocletian and his division of things into dioceses also helped, but I think once they started down the path of dual Augustus/Caesar rule it was bound to fail due to the inevitable clashes of personality.
