What you mean by "civilizational gap" is a bit vague. Are you referring to "complexity"--that is, the range of specialization, the power of leaders, the extent of long distance trade, etc.?

I'm not formally trained as a historian in late antiquity and the early Middle Ages, but I've done considerable research on the topic of migration and the relation between Rome and its Germanic neighbors (it's my second great love, second only to the First World War). So here's my attempt to answer your question.

I recommend checking out Peter Heather's "Empires and Barbarians." He deals with the issue of relative complexity between the Roman Empire and its "barbarian" neighbors. He argues that less complex societies develop greater complexity when they are exposed to and interact with a comparatively more complex state. Germanic peoples of the first century AD, for example, tended to organize themselves into small units and on their own were incapable of challenging Rome. Germanic leaders had little power to coerce followers into, say, fighting, and groups engaged in less long-distance trade. By the late fourth century AD, however, groups like the Goths, Alemanni, and Vandals formed larger confederations of tribes that were much more militarily capable. They engaged in long-distance trade with Rome and their other Germanic neighbors. Their populations had also grown considerably.

Moreover, the Germanic groups that settled the former empire quickly adopted Roman and Latin customs. Theoderic the Great, the Ostrogothic king who invaded Italy in the late fifth century, presented himself as a Roman emperor. In fact, this image of Theoderic so closely resembles a Roman emperor that it was long assumed to depict Justinian.

So, yes, I think the difference in complexity between the Germanic peoples and Rome is somewhat exaggerated. Rome was certainly the more complex society by any of the measures I noted above, but the Germanic confederations had grown in complexity a great deal between the first and late fourth centuries AD. Moreover, Germanic leaders wanted to be part of what Rome had to offer. Indeed, this was one of the driving factors that led them to move into Roman territory (that and probably the Huns). They thus understood what Rome meant and what it meant to be Roman.

Hope that answers your question!

Sources: Peter Heather, The Fall of Rome; Peter Heather, Empires and Barbarians; Peter Heather, The Goths; Halsall, Barbarian Migration and the Roman West

Edit: I'd just like to add to that idea of the brutish Germanic tribesman hellbent on conquest and plunder is a remnant of the Renaissance thinking about the "dark ages." These were complex societies that, above all, wanted to share in Rome's prosperity and material wealth.