Counting in vigesimal, as French does (in France and Quebec) was at one time common enough in much of Europe; compare the English score ("three score and ten") and modern Danish usage where 50 is halvtreds meaning "half of the third (twenty)". Base-20 currencies - variations on the old Roman 1 librum = 20 sestertii = 240 denarii - were commonly used, the last surviving being British and Irish pounds, shillings and pence which made it as far as 1971. The French replaced their livre, sou and denier with the franc, of 100 centimes amidst the rationalisation of systems of measurement that followed the Revolution which also introduced the metric system and, less successfully, the revolutionary calendar with its ten-day weeks.
At this point the decimal words for 70-90, "septante", "octante" and "nonante", which echo the forms used in Latin and other modern Romance languages, were already in academic use; the dictionary of the Académie française of 1694 remarks that huitante or octante are "scarcely used in common parlance ... but used in arithmetical calculations"1, but it appears that they were taken up seriously as being the way forward, and teachers were instructed to use them all over the country. However, there was strong resistance in Paris, and the decimal forms never really took root there, although they did in the provinces; France being an intensely centralized society, however, what goes in Paris is what goes, and the decimal number system faded out in French use during the 20th century, although it was still strong in the south-east until relatively recently (the Occitan language and many regional dialects used Latin-derived base-ten forms as well, which helped). It is still generally understood across France, but will earn you an extra condescending look.
In other parts of the French-speaking world, outside the sway of Paris, things were different. Quebec, ruled by the British since well before the revolution, and latterly the fiercest bastion of francophone linguistic puricism, had never been exposed to the decimal forms, but in Belgium and Switzerland, they stuck; here too, Wallon and the dialects of the Suisse Romande had never used base-20 forms, and the state saw no reason to retreat from the more "modern" forms. As a result, in those countries, septante and nonante (and derived forms) are normally used, while soixante-dix and quatre-vingt-dix are considered odd variants. This is not colloquial, slang or dialect (at least in the pejorative sense) usage: my son's Belgian birth certificate starts "Le trois avril mil neuf cent nonante-six, à huit heures quarante-sept minutes, est né à Uccle, ...". However, in Flanders I have heard the héxagonale forms being ostentatiously used in a clear attempt to indicate that the speakers were merely doing a favour for French visitors and had nothing to do with francophone Belgium.
In francophone Belgium, "quatre-vingt" is still used for 80, but only in base ten compounds (ie up to quatre-vingt-neuf); some dialects of Walloon use "ûtante". In Switzerland the preferred form is "huitante", the archaic form "octante" having fallen out of use for "at least 50 years" according to themanwho.
Elsewhere in the French-speaking world, it appears that the base-20 forms are, as you might expect, used in the former French colonies in Africa and elsewhere; the former Belgian colonies of the Democratic Republic of Congo and Rwanda are reputed to follow the Belgian pattern as you would expect, but they do not have a heavy web presence and I could not find any instances of "septante" or "nonante" in the .cd domain (and I did find one "soixante-douze"). I am also seeking enlightenment as to what is used in the Valle d'Aosta, the partly French-speaking corner of Italy, where it is rumoured that "octante" may still be extant, likely with help from the Italian cognate "ottanta".
Also referred to:
Le Robert Dictionnaire historique de la langue française
With thanks to mkb for asking, and to Mrs Heathcote, the Buckinghamshire French teacher who gave the then ten-year-old Herring a telling off in front of the whole class for pointing out these differences in a test.