The front squat is an excellent movement to get bigger, stronger quads - probably the best exercise there is. Because you're position is more upright, it emphasizes the quads more.

 

Because you're holding the weight in front of your body, and you must remain upright your entire core gets a great workout trying to stabilize the weight and keep you upright. If you find you're leaning forward a lot when you back squat, you'll quickly get stronger and fix this problem if you switch to front squats for a while.

 

Because the bar is placed in front of you, on your shoulders, the front squat is a lot like other exercises and strengthens the "base" of those lifts/presses. Power Cleans, Overhead Presses, Overhead Squats, etc. all share the same "starting" position as the front squat. Getting stronger and more comfortable with Front Squats will help you in those exercises too.

 

You're more upright with the front squat. Your spine is in a more neutral "straight" position. This combined with the fact you're lifting less weight is healthier for your spine in the long run.

 

Most people have lower back problems and back squats aggravate these conditions. But because your upper body is mostly upright (instead of leaning forward like in a back squat) there's less force on your lower back.