Did the system of feudalism function differently than Europe had functioned during the Roman Empire?
The above answer is only partially correct. The Roman system of government was based on the rule of law, beginning with the Twelve Tables and later by decrees issued by the ruling Emperor. The law was written down and commonly known and understood. The Feudal system was based on the old Germanic principle of loyalty. The Germanic tribes who ultimately migrated into and occupied the Empire had no written language, accordingly loyalty and the reliability of ones word was vital. Under the Roman system, one conducted oneself in accordance with the applicable law; under the Feudal system, one conducted oneself according to ones pledge of loyalty, quite often expressed in the ceremony of homage and fealty.
Other than the lack of a commonly understood written system of laws, Europe also was characterized by a lack of centrality after the collapse of the Roman Empire. The Roman government and its system of roads made its rule relatively uniform (even though the size of the empire made this uniformity difficult and ultimately led to its demise.) Following the collapse of the Empire and particularly during the period of Viking and Magyar invasions, centrality was impractical. It was necessary for defense (and therefore government) to be localized. The entire Feudal system was based on the importance of loyalty and locality of defense, a situation which did not exist during Roman times.
If one compares Feudal Europe to the Byzantine Empire which was still Roman in its customs and traditions, the distinction becomes clear.
Yes, it did.
During Roman times, the Roman part of Europe was ruled by a centralized government, not by a web of feudal relationships. The Roman Empire's central ruler was in Rome. Under him were various governors and other officials who governed in the provinces according to Roman laws and to the policies set out by the emperor. This was a system somewhat like ours today where the various people holding office were civil servants of a sort, not feudal lords who actually ruled their domains in whatever way they liked.
Roman times were much more organized and centralized. There were, at times, civil wars because of the power of major army leaders. But there was not this system where members of the nobility had their own little fiefs that they ran however they liked. Instead, there was a clear system of top-down rule where the whole empire was under the command of Rome.