Question

how did religion affect the Byzantine separation from rome?

Answer 1

The Byzantine empire's religion was Orthodox Christianity while the Roman empire's religion was the Roman Catholic Church.

Some differences between the two religions that helped cause the split were:
The Iconoclast Controversy
The use of leavened instead of unleavened bread in worship
They disagreed on proper interpretations of the bible
The power of the Pope
and whether Constantinople had equal importance as a Christian city as Rome was.

However, the Great Schism is the actual event in which the two churches split.

Got it? =)

Answer 2

Great answer by ilovemacncheese. That's pretty much what happened between the East and the West. The East had one view of Christianity while the West, under the influence of Rome, went another way.