The Gauls were a specific Celtic tribe, or possibly group of tribes. All Gauls were Celts, but not all Celts were Gauls.
As for what the Franks were, I quote the opening sentence to the article on them in the Microsoft Encarta Encyclopedia, 2002 edition: "
Franks, group of
Germanic tribes that, about the middle of the 3rd century AD, dwelt along the middle of the lower Rhine river." (Italics mine.)
And my comments were not about Teutoburger Wald. I was saying that, yes, the Germanic peoples destroyed the Western Roman Empire, but once civilization in that part of the world recovered from the blow, it was the Roman civilization, albeit in new forms, that was dominant. The Germanic peoples eventually forgot their old ways and took on Roman-flavored systems, and I was using Charlemagne as a piece of evidence; he was a Germanic ruler who considered himself Emperor of the Romans, as did all his successors in what eventually became the Holy Roman Empire.
As another piece of evidence, I could say that though the modern English--and therefore the Founding Fathers of the U.S.--are descended more from Germanic peoples (Anglo-Saxons and Normans, the latter being a type of French-speaking Viking) than they are from Romans, our governmental and legal systems are nevertheless derived from those of the Romans. We've lost our old Germanic ways and now live under a system derived from the ones the Romans created. (I say
we lost
our Germanic ways because I personally am descended from the Germanic barbarians, and not from Romans. So I am one of the ones whose ancestors lost their ways and took on the Roman, even though the Romans themselves never conquered them.)
But perhaps I have abandoned the scope of the original question, and pontificated at too great a length on something irrelevant. If so, then I apologize. But sas has already clarified that the question was not about Teutoburger Wald, but about the barbarian invasions later on, so I don't really think that I have.