Actually, let's take a look at the differences between the two, and how those differences brought the two parts of the old Roman Empire into conflict.
The Byzantine empire was largely shaped by its Greek heritage.
At the same time, the west was influenced by Germanic and Frankish cultures.
The most important conflict between the east and the west was most likely the one between the churches, because as I mentioned a bit earlier, culture and religion played a significant part in daily life.
The first major disagreement concerned religious icons, and was about the fact that the Byzantine Empire banned them when many people were using them. The ban was eventually lifted but not before the destruction of much religous art.
Another problem occured when Pope Leo III decided that it was not fit for the empress of the Byzantines to be ruler because she was female, and so crowned Charlemange, the king of the Franks, as the Holy Roman emperor. You can most likely see how it upset the Byzantines.
...And this is the story of how the Roman empire split apart.