There are advantages to both forms of racing. In autocrossing, you can drive a street legal car. There are some restrictions, for example, don't expect to race the family minivan, but you can race almost any car. There are several different classes, and your car would be put into a class with others of its type, to make the racing fair for everyone. In autocrossing, however, you are only racing against the clock.
Cars compete for fast times on a course laid out with cones.
This is so that you won't have to worry about crashing into concrete walls, or other competitors, with the car you want to drive to work tomorrow. But if you really want to race with other competitors, on the same track, with passing, then look into karting. Now this may not seem so bad, but you also need to have a way to get the kart to the track, as well as store it between races. For all forms of motorsports, you will need to buy protective gear. For autocrossing you will need DOT or Snell 95 Helmet.