As the name implies, the main difference between front squats and the classic squat is that you place the barbell in front of your neck instead of behind it. This slightly different angle has some additional benefits to the classic squat.
#1 More quads
Keeping the bar in front of you shifts the center point of gravity noticeably, forcing you to maintain a much more upright position as you squat down. This in turn puts a lot more emphasis on the quadriceps muscle (front thigh) compared to classic squats, which bring in more hips, butt and hamstrings (rear legs) into play.
#2 Safer back
When people get tired doing classic squats, they tend to sag forward a little, rounding their backs. This is an exceptionally bad idea, especially with a loaded barbell balancing on the shoulders. With front squats it's physically impossible, as you'd do an instant faceplant on the floor.
#3 Better form
This is merely another side of the same coin outlined before; you can't cheat, period. The inability to cheat makes better progress automatic, tying into the first argument to boot. One drawback is that you'll not be able to handle the kind of poundage you did with classic squats, which may be something of an annoyance. Don't let it get to you -- check the ego at the door and remember what'll give you best results in the long haul.
Finally, the mere fact that it's a new way of training quads is a potent benefit of front squats all by itself. Muscles thrive on variation; the more you surprise them each week, the stronger they tend to get. Conversely, if you stick to a predictable routine, your muscles have little incentive to improve once they have achieved that level of fitness. Thus, mixing classic and front squats with their different poundages and strictness tends to give the best overall results compared to only one kind of either exercise.
Front squats are excellent for beginners and experienced weight lifters alike looking to build strong legs in a safe but effective manner. Just don't forget to stretch thoroughly afterwards, as the soreness can be brutal if you haven't done front squats before.