uhh...well depends on the time period. around 1050 and later, the Byzantine empire was in its day, but the Roman empire didnt even really "exist" because it was fragmented into other kingdoms and wasnt really...an empire. the Byzantine empire did exist, and Alexius Comnenus was the emperor of the Byzantine empire at around 1095 or so (around the time of the first Crusade).. . the Byzantine empire around the crusade was mainly Greek Orthodox, whereas what was fragmented Roman empire was Roman Catholic. obviously, both were Christian, but if i remember correctly the Byzantine empire didnt recognize the pope as the head of the church.