There isn't an enormous amount of variation, but there's some.
First off, as you've noted, every class can equip all weapons and armor. Every class can also make use of the various 'fart' spells that you'll learn throughout the game.
The main difference comes in the special abilities available to each class through the use of Power Points, and the upgrades to those abilities that can be had. For example, the Jew has access to a number of debuff centric and AoE attacks, including an ability that cuts a single target's armor in half, and an AoE ability that can apply Attack Down, Bleeding, Burning, and Grossed Out. Thieves have the ability to apply large numbers of stacks of Bleeding. Mages can apply Burning and Slow with 'elemental' attacks. etc. There isn't really a 'strongest' class, or even a real set of pros/cons here; it's mostly just flavor, though different movesets do have different 'QTE' triggers for bonus damage or debuffs which you may find easier or harder to time and/or activate.
You can review the various skill trees at the Official Stick of Truth Website and play around with various 'builds'. Note that the 'Perks' available (for finding more friends) are the same for all classes, though some fit the abilities of some classes more than others. Abilities, and their various upgrades are literally the only difference here.
The other major difference is that at a few points in the game, you'll get class-specific armor with bonuses tailored to your class and a look that strongly fits that theme. Obviously the Jew has the most distinctive looking sets here, but I found the bonuses - increases to damage when the New Kid is debuffed - to be really underwhelming. YMMV.
There are no advantages to using specific armor or weapons for specific classes, except insofar as those armor or weapons synergize with your class abilities. For example, items that provide a bonus to fire damage can be very strong for Mages, but they can also be strong for anyone else with other equipment setups or perk/upgrade choices.
Finally, as a minor difference, folks around South Park will occasionally react to you based on your choice - or at least, in the case of playing a Jew, folks will frequently recommend you get a nose job. Cartman will also make snide and vaguely racist comments about your selection based upon your ethnic selection at character creation. For example, he will call out white thieves as being unusual, black fighters as 'scary', and 'Magicians from Jersey' as... something.
There is one other difference. There's an achievement for getting Jesus to be your Facebook friend as a Jew. However, you can do this immediately after selecting a class and completing the tutorial, by simply walking out of Cartman's house, heading over to the Town Hall, speaking to Father Maxi, and then heading to the Church and Finding Jesus. Since you can do this achievement without really playing through the game with any degree of significance (It's literally 10 minutes worth of a reload if you'd prefer to play a different class), I hesitate to even call it an 'advantage' of choosing the class.