Crunches are where you contract your abs and simply raise your torso and shoulders just off the ground, like Amsterman posted. These work your abs over a much smaller range of motion so less muscle is activated and without your feet being anchored you can't use your hip flexor to help you.
At the end of it they're both working towards the same goals and the exercises share many similarities. In the bodybuilding community situps are reportedly able to build more muscle mass than crunches (when done within the correct rep range people, don't rape me) because they have a much greater range of movement and so, recruit more muscle fibres, potentially leading to greater muscle growth. Also situps can be more beneficial to people who do high impact activites because the hip flexors can act as shock absorbers, so working them can help to protect your body from the rigors of running, jumping and the like.