What are the differences between a DC motor and an AC motor?

The most basic difference is the power source. A.C. motors are powered from alternating current (A.C.) while D.C. motors are powered from direct current (D.C.), such as batteries, D.C. power supplies or an AC-to-DC power converter. D.C wound field motors are constructed with brushes and a commutator, which add to the maintenance, limit the speed and usually reduce the life expectancy of brushed D.C. motors. A.C. induction motors do not use brushes; they are very rugged and have long life expectancy.

Internally DC motor also needs alternating current. The supply which is DC is converted to AC by the commutator. Also in case of DC generator, the internally generated emf is AC, but the output is DC because of commutator which is also called mechanical rectifier in case of DC generator. 
In AC motors as the supply is AC, there is no commutator present.

DC motor is the rotating motor that is capable of converting DC energy into mechanical energy (DC motor) or converting mechanical energy into DC energy.
AC motor is used to implement the AC power and mechanical energy conversion. Due to the great development of Veichi AC power systems, AC motors has become the most commonly used motors.