Question

Which conflict was harmful to the Byzantine empire but not to the Roman Empire

Answer 1

Third option ("religious disputes between christian sects") is the most accurate. The Byzantine Empire was severely affected by the conflicts with the Catholic potencies of western europe resulting from their doctrinarian differences after the Great Schism of 1054.

From then onwards Byzantium followed Orthodox Christianity, which was seen as a heretic religion by western european powers and helped them justify wars against the empire, such as the Fourth Crusade and the sack of Constantinople in 1204, leaving it vulnerable to invasions by the incoming turkic tribes of the east that would later form the Ottoman Empire.