Car wheels have treading on the outside of each tire to increase friction between the car and the ground. Unfortunately, if the treading is worn out during the winter time the effectiveness is limited. Due to a lack of friction between the ground and the tire, and Newton's first law of motion, the car slides on the ice.
The reason that the treading prevents slipping in the muddy and snowy conditions is because of the grooves on the wheel where the mud/snow can get into. The muddier or snowier it is, the wider and deeper the treads should be. As the wheel turns the grooves dig into the soft material of snow or mud, allowing the wheel to rotate with friction to cause the entire car to move.
Without the treads the car wheel would still rotate from the torque applied to the wheel from the gas of the car, but it would not move forward because there would be no friction between the car and the ground to cause the wheel to roll.
When there is no snow the chances of slipping are less because pavement has a higher coefficient of friction than ice and snow.