This might be a disciplinary problem, and perhaps a bit of a political one as well. That is, it's my impression that people who study the early middle ages tend to refer to the Franks. The term suggests an emphasis on the fall of Rome and the Germanic background of the tribe, although these are both concepts that modern historians vehemently debate. Scholars of the later middle ages, however, tend to refer to the French, which has connotations of setting up the aristocracy of the ancien regime and connects to modern notions of nationalism.

In the 1800s and early 1900s, there was a substantial debate about whether modern French civilization (and implicitly the population as well) descended from Germanic Franks or Gallo-Romans. The way we use these terms today still carries echoes of this old debate. Discussion about the Franks resonates with the concerns of bygone German nationalists who wanted to emphasize how new Germanic tribes had moved in and conquered Rome, creating modern European civilization. Discussion about the French in contrast suggests that French civilization emerged after this period and therefore is not indebted to or derivative from a Germanic culture.

After two world wars that have pitted German territorial claims over the French, scholars are a lot more careful about how they use and discuss these periods and associated terminology. Nevertheless, we're still often responding to arguments put forward by earlier generations of scholars, so there's a lot of inertia behind our continued use of these terms.

Perhaps one good continuing justification for this division between "Frankish" and "French" terminology might be that early medieval historians tend to rely more on Latin documents (i.e. referring to the gens francorum) whereas late medieval historians are more likely to encounter vernacular documents talking about les francais. I'm not an expert of the later middle ages, however, so I can't really say when that begins. Perhaps one of our high medievalists will pitch in their perspective.

At any rate, unless you've got something specific in mind that separates Frankish from French, I suspect this is much more a question about how we do and discuss history, rather than a question about historical events themselves. I don't think Charlemagne or Hugh Capet woke up one morning and said, 'Okay boys, today we're French.' Periodization and terminology tend to be problems that historians make for themselves, rather than problems that people living in the past actually had to deal with and describe.