At the third Council of Tours, the priests were ordered to preach in vulgar Latin, as the common folk could no longer understand formal Latin. This was in 813 CE. While it may still have been Latin in name, this language would likely have been significantly different from the VL at the fall of the Western Roman Empire.

Vulgar Latin, particularly in outlying regions of the former Roman Empire tended to be influenced by the local languages, as is usual. For instance, the formal Latin equus for horse was replaced by caballus, a word originating in the Gaulish tongue.

The other major factor that would have changed in the intervening centuries between the fall of the Western Roman Empire and the Council of Tours is pronunciation. Old French in particular was guilty of significant vowel breaking - where a monopthong is broken into a dipthong or further e.g. the Latin short /e/ in mel, became /ie/ miel, or honey. Of the original dozen or so vowel sounds found in Latin, all were changed.

Most significantly, vulgar Latin, and subsequently Old French placed more emphasis on stressed syllables, which tended to shift pronunciation and resulted in the now standard dropping of final letters when pronouncing French words. This shift lowered the morpheme:word ratio of OF/VL due to the end-conjugation of many Latin words. This meant that OF/VL relied significantly more on context that did formal Latin.

From all this we have a date of certainty (813 CE), but indications exist that the loss of mutual intelligibility could have occurred significantly earlier, as language undergoes speciation in much the same way that living organisms do, on a continuum.

EDIT: You might consider posting this same question over in /r/linguistics for a more in-depth response.