In which way did spoken French change? To the extent of the Great Vowel shift?
There have been changes in both pronunciation and more generally in the way people use the language.
The R sound has changed from a trilled sound similar to Italian/Spanish to the uvular sound similar to German. This process is just about complete in Europe but is ongoing in Canada and Africa, where you can hear both R sounds.
The -oi has changed dramatically:
moi used to be pronounced like "moué" and now is "mwa". This change is ongoing in Canada : the "moué" pronunciation still exists but is becoming stigmatized. So the French word
royal was once pronounced very similarly to the Spanish
real but now is totally different. (Interestingly, there is the city of Montréal (an archaic name) but the mountain in the middle of it is called Mont-Royal, which is modern.)
French used to have numerous diphtongues, which helps to explain a lot of spellings. These are all gone now from European French though some survive in Canada. Likewise,
a and
â traditionally designate two different "a" sounds (closed and open) but this also is disappearing in France (though less so elsewhere).
Regarding usage, some verb tenses (most notably the passé simple, the French form of the preterite) have just dropped out of use in speaking. The passé simple is still used in literature but would be considered strange and pompous to use in speech. The passé composé tense (which used to only serve as the present perfect) has replaced it nearly 100% of the time.
Finally, whereas negation in French originally occurred with just "ne" before a verb, and then (by Molière's time) had become a pair (ne . . . pas/point), in the spoken language nowadays it's very common to drop the "ne" and just use the second part (although in writing, both parts are still used). So "I don't like" was originally "Je n'aime" and then became "Je n'aime pas" and now (in speaking) is frequently "J'aime pas".