Probably iconoclasm which caused a split between east and west around AD 680-850. Second most important difference was language. The western church conducted it's business in Latin and the eastern one in Greek. Political differences, too, between the Byzantine Emperors and the Frankish Emperors contributed.
You are asking to a simple answer to something that is intimately bound up with some thousand years of history.
The Byzantine Empire's dates run from  A.D. 330 – 1453. Until the Eastern Orthodox Churches split from the Catholic Church in A.D. 1054 there were no "branches" of Christianity, there was only the Catholic Church.
Catholic
A rift developed between Latin Christianity and Greek Christianity which then led to the schism between the Catholic and Orthodox Churches. The conflict between the two led to mutual distrust . The Orthodox Church was the church of the Byzantine Empire.
Albanian Byzantine Catholic Church Armenian Catholic Church Bulgarian Greek Catholic Church Chaldean Catholic Church Coptic Catholic Church Patriarchate Ethiopian Catholic Church Byzantine Church of Croatia, Serbia and Montenegro Greek Byzantine Catholic Church Hungarian Byzantine Catholic Church Italo-Albanian Byzantine Catholic Church Macedonian Catholic Church Maronite Catholic Church Melkite Greek-Catholic Church Romanian Greek-Catholic Church Ruthenian Byzantine Catholic Church Slovak Byzantine Catholic Church Syriac Catholic Church Patriarchate Syro-Malabar Catholic Church Syro-Malankara Catholic Church Ukrainian Greek-Catholic Church
Greek Byzantine Catholic Church was created in 1907.
Byzantine Catholic Eparchy of Parma was created in 1969.
Byzantine Catholic Eparchy of Phoenix was created in 1982.
Byzantine Catholic Eparchy of Passaic was created in 1963.
Byzantine Catholic Archeparchy of Pittsburgh's population is 58,997.
St. Nicholas Byzantine Catholic Church was created in 1918.
Byzantine Catholic Archeparchy of Pittsburgh was created on 1924-05-08.
St. John the Baptist Byzantine Catholic Cemetery was created in 1923.