Let's start with tire pressure. A car tire has somewhere between 2 and 3.5 bars (that's 29 to 51 psi) while airplane tires go up to 24 bars (350 psi). Next, they must withstand extremely heavy loads for a short time. The spinup and impact load of a standard airliner landing would let any car tire burst in the same circumstances.
In order to be retractable, and even more when not retractable, airplane tires must be as small as possible. An extreme example is the Tu-144, where each main gear had 8 wheels to keep individual tire size down.
The tread on an aircraft tire does not have to transmit torque continuously, only during spinup and when braking. It must give good directional stability in crosswind and avoid hydroplaning as much as possible. This is best achieved with circumferential grooves.
The number of plies of an aircraft tire are more numerous and the rubber thickness is much higher than those of a car tire, proportional to the tire pressure and the spinup and braking loads.
An important addition comes from JohnK:
An important design factor unique to aircraft wheels, even GA wheels, is that the tire must be unable to come off the rim under extreme lateral load with underinflation, which means much more rim/bead overlap, which in turn means the tire can't be stretched over the rim to install it, and the rim is instead made in two halves bolted together.
The tires also need to be able to withstand far more heat than a normal car tire. One of the requirements for a jet aircraft to be certified by the FAA is that a plane with minimum specification brakes must be able to fully reject a takeoff using only the brakes. The plane must then still be able to taxi for five minutes (worst-case assumption for an airport with no on-premises fire/rescue), which means the tires need to be able to withstand that extreme environment for a short time.