By the fourth century CE, the Roman Empire had become too big for its own good. So Emperor Diocletian came up with a strategically simple solution--split it in two. To the east is Byzantium, capitol Constantinople. To the west is Rome, capitol Rome. The latter was the target of a barbarian invasion that succeeded in reducing the glory of Rome into the Dark Ages. While Rome endured centuries of war, famine and illiteracy, Byzantium still adulated in luxury. In fact, it was Emperor Justinian who proposed reintegrating the two back into one new Roman Empire. (But certain circumstances strangled that ambition before it could get off the surface.)
I am merely curious as to what would happen if it were the other way around--that the sacking happened in Constantinople, not Rome. It was a much bigger city, with more money and possibly more food. Assuming that some barbarian army, be it Germanic, Slavic or even Hunnic, invaded and destroyed Constantinople, would the course of history be noticeably altered in any way?