With respect to targeting this achievement, there isn't a huge amount of difference between the classes.
The one thing that may significantly impact your quest for this achievement is the fact that if your buddy dies, if you can revive them before the end of the fight, you're still on track for the achievement. What that means is that you may want to build your character and select skills in such a way as to minimize your use of debuffs or Damage Over Time effects. (This has the side effect, incidentally, of making much of the game much, much, harder, for what it's worth, so I'm not sure I recommend going all in on the strategy - easier to reload, IMO.) This will allow you to make combat a bit more predictable, at the expense of it lasting longer. Incidentally, the class best suited to avoiding the application of DoT's - the Fighter - is also the class with access to the widest variety of abilities that can lower your opponents Attack, which is also useful for pursuing this achievement.
Another option is to focus more on preventing foes from acting. In this case, the Mage has access to a Stun from right near the beginning of the game, and later, gains access to the only innate Slow effect - which can be very useful.
Finally, the Jew and Thief both have access to abilities that can apply 'Pissed Off' to targets; this functions as a taunt of sorts, forcing enemies to attack your character instead of your Buddy. (Do note, however, that the Jew gains access to this at a much lower level than the thief).
Really though, all of these differences are pretty minor. What matters more is your consumable use, and willingness to reload if things do go a little pear shaped. Every class has the ability to do lots of damage, and once you get to a high enough level to kill a mob in one attack, (the Sucker Punch perk helps) the Swat Helmet and some smart use of your 'Shoot' skill on the map will mean that you can end many fights before an enemy even has a chance to try to hit your buddy.
Best Answer
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.