Besides salary, is there any difference between "classes" of faculty in France?
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.