The Byzantine Empire is technically a surviving part of the remaining Roman Empire. When the Roman Empire split, the Byzantine Empire managed to last another thousand years. It had its own Orthodox Church. Western Europe was basically dominated by the Pope of the Roman Catholic Church - his word was law. People thought that the Pope knew what God wanted, so whatever the Pope said, the people believed. Both Western Europe and The Byzantine Empire were dominantly Christian, though they did not agree on some things. Both of them traded on the old Silk Road.
The Byzantine Empire was centralized and retained a strong bureaucratic system, whereas Medieval Western Europe was decentralized under feudalism. Economically, Byzantium thrived on trade with a monetized economy, while Western Europe transitioned to a more agrarian, barter-based system. Religiously, the two empires were dominated by different forms of Christianity, with the Byzantine Empire following Eastern Orthodoxy and Western Europe adhering to Roman Catholicism, leading to significant theological disagreements.
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