Besides salary, is there any difference between "classes" of faculty in France? While learning about the different faculty jobs in France, I discovered that the two different ranks of faculty are subdivided in classes: maitre de conférence are sorted in "normal class" and "outstanding class", while professors are sorted in "second class", "first class", "exceptional class". You start in the lowest class and if you're good you can be promoted.

From an outsider perspective I could not find any difference other than salary. However reading the history of these ranks they used to correspond to, I think:


maitre-assistant -> new maitre de conférence
old maitre de conférence -> professor "second class"
non-titular professor -> professor "first class"
professor titular of a chair -> professor "exceptional class"


Today, other than salary, does the class change anything? Does a higher class faculty have more weight with the university or something? More responsibility? Or is it purely financial?
Generally speaking, it is financial. The responsibility difference is between HDR-holders and non HDR-holders (HDR = Habilitation à diriger les recherches). 

Maîtres de conférences are not always allowed to supervise research alone (for example: you cannot supervise a PHD student alone). 

To do so you need to hold an HDR: it is a university diploma, something like a smaller PhD thesis that proves that you can supervise others. It's similar to Habilitation in many European countries, like Germany for example. 

All "Professeurs" have an HDR : it is a condition (but not the only one) to get promoted. Some "Maitre de Conférences" have an HDR, some do not. 

Please note there are two fields in French universities where the "Maître de Conférences/Professeur" distinction is different, for historic reasons : Law and Economics. You become a "Professeur" by completing a competitive examination: "l'agrégation". HDR is this case is not necessary.