Question

How does this controversy relate to the development of the Byzantine Empire?

Answer 1

Byzantium or Constantinople? A Second Rome or a Hellenistic Empire? Greek or Latin? Orthodox or Catholic?
Constantine founded the city named after him as the Second Rome. Byzantium is derived from the original fishing village. He intended it to be a mirror of the first at a time when the Empire had become too big for one man. The problem was that the Eastern Empire was stubbornly Hellenistic in culture, Greek was the lingua franca not Latin the official language, and frankly the East rather thought the West somewhat uncivilised.
Even in the Church certain divisions persisted. Greek was more subtle than Latin and some formulations regarding the Trinity and Christ's divinity were more nuanced in the Orthodox Church. The Pope claimed primacy over the whole Church, but it hardly suited the Eastern Emperor to have a Western Pope, out of his reach, lay down Church Law. The Patriarch demurred.
The old Rome collapses under the pressure of barbarian invasions, despite intervention from the Eastern Empire. The Pope gains an independence largely due to the lack of secular authority which affects the future history of Catholicism. The new barbarian leaders harbour ambitions to recreate the Western Empire in their own image.