The history of Rome and the papacy is quite long and complex. Though it hadn't been the capital of the empire for about a century before the empire fell (the capital was Ravenna), Rome was still considered an important and rather big city of the empire. The senate still existed but the bishop (the pope) was gaining more and more influence over the affairs of the city, as it was common in early middle ages Europe. Rome did get smaller after the greek gothic war and the plague, and stayed for more or less two centuries under the control of the byzantine empire. Byzantium had the more pressing matter of the eastern border so they didn't managed to defend Italy when the Lombards invaded and conquered more or less half of the penisula startinf from 538 A.D.. Rome and other territories managed to stay under byzantine control but had to face the constant threat of a Lombard invasion, with Byzantium not always able to aid since it has the more pressing matter of the eastern border and infighting, so the pope ended up as the authority in the city. The byzantine control lasted until the VIII century, when the Lombards managed to conquer lots of the territories under byzantine control. King Liutprand managed to conquer Sutri (near Rome) but eventually decided to donate the territory to the pope in 728 (Donation of Sutry), the first accolade of sovereign territory to the papacy. It was maybe under pope Paul I (757-767) that one of thr most important false documents of history was redacted : the donation of Constantine is a forged roman decree according to which the roman emperor Constantine the Great left the territories of the western roman empire to the pope. This fake decree is a symbol of what the papal strategy had become: they consider themselves heirs to the roman empire therefore they should control the byzantine territories in Italy. This new strategy wasn't particularly appreciated by the Lombards or Byzantium. However Byzantium was deep into the iconoclastic war so it couldn't exert any kind of meaningful influence in Italy. As for the Lombards, the Pope Stephen II decided to ask the french King Pepin the Short (Charlemagne's father) for help, starting the very long and complex relationships between the papacy and the french crown, not to mention the long battle for primacy between the Pope and the emperor of the Holy Roman Empire, since Charlemagne has been crowned by the pope but since Charlemagne's successor Louis the Pious the pope had to swear loyalty to the frankish emperor. This is a very simplified response that regards mainly Rome's history in the high middle ages . As you can see Rome has never really lost any importance but a mix of opportunity like a weakened central power like Byzantium, riddled by infighting, iconoclasm, constant war on the eastern border with the Persian empire and then the Caliphate, and also a good foreign policy like an alliance with the rising Frankish power allowed Rome to make a comeback and stayed one of the most important cities of Europe and one of the cornerstone of Italian history with a long lasting influence. The Vatican is still a sovereign state after 2000 years.