I often think about turmoil in countries like the former Soviet Union, The Egyptian Empire, the Qing Dynasty, these great empires. But when they were close to falling did the general population of the areas have any clue the empire would actually no long exist? Like waking up and realizing today we don't have to pay taxes to the Soviet Union....?
The issue with the Fall of the Roman Empire in the west was that there was no one definitive date where everyone sat down and realized they were no longer Romans. Indeed for most of the Roman World, their empire didn't fall for another 1000 years, by this point in time the Eastern half of the empire had been more economically, militarily, and political powerful for quite some time, and the Carolingian Empire and the later Holy Roman Empire both claimed to be the legitimate successor to the Roman Empire until the early modern period.
As for the Romans who did live in the west there was no cataclysmic event that severed the ties they had to the "old" Roman Empire. The Barbarian kings styled themselves as Roman rulers more often than not and claimed to still be continuing the Roman Empire just with a new batch of elites in charge. The situation was made all the more complicated by the version of Christianity that was practiced which essentially defined the Roman Empire as all of Christendom. (I am paraphrasing from Jas Elsner's Imperial Rome and Christian Triumph)
On top of that the Byzantine Empire made numerous forays into the West, notably into Italy and Africa under Belisarius and Justinian I, as well as into the old Iberian provinces.
In short there was no definitive fall of the Roman Empire in the west due to a variety of factors.
For further reading I would suggest the Elsner book I metioned above. Peter Heather's Empire and Barbarians and Peter Brown's The World of Late Antiquity