Rome did not fall in 476 CE
The thing about the decline of a massive empire is that you cannot pinpoint the time at which it ceases to exist. You can pinpoint when it has ceased to exist, but determining when it fell is much more inexact endeavor.
The fall of the Roman Empire was a process which took place over the course of centuries.
First, we must remember that the Roman Empire was divided into an Eastern Empire and a Western Empire in the fourth century. The Byzantine Empire (the Eastern Roman Empire) endured into the fifteenth century until Constantinople fell to the Ottomans. So in discussing the fall of the Roman Empire, we must remember that we are only discussing the Western Roman Empire.
Here is a map showing the East/West division (please note that this map shows the height of the Empires; within the fifth century time-frame being discussed in this post, the West controlled a significantly smaller amount of territory than is pictured below):
By 476, the Western Roman Empire had, for all intents and purposes, already fallen. Yet at the same time, it would continue to live on for centuries.
What happened in that year was the deposition of the last traditional emperor of Rome, Romulus Augustus, by Flavius Odoacer (historians are unsure as to whether he was a Goth or a Hun). That’s all. And it was hardly a tumultuous event; the peoples the Romans referred to as “barbarians” had, in fact, been major political players in Rome for over 100 years prior to the date of the perceived fall. Many of those tribes were fully assimilated into Roman society, and it was often the case that they functioned as the true powers behind the imperial throne. We can only assume that Odoacer grew tired of the charade and decided to make it official.
By the fifth century, these tribes—the Vandals, the Huns, the Ostrogoths, the Visigoths, the Lombards, the Franks—had assimilated to the Roman way of life and were exporting it to areas outside the bounds of the Empire. Western Roman culture outlived the Western Roman Empire as a result of these tribes.
For at least two centuries after the generally accepted time of the fall of the empire, the Roman culture lived on. However, by around time of Charlemagne (late eighth, early ninth century), most of the populace had come to view the cultural heritage of the Roman Empire with suspicion and disdain. Thus, we can probably say that the Western Roman Empire had ceased to exist by the mid or late seventh century.