What advantage does an AC motor have over a DC motor?
DC motors are also used where controlling speed and torque are important. Also the starter motor in a car runs off a (DC) battery and the DC motors that operate windows and seat positions in a car are small and powerful because they use compact powerful permanent magnets. The battery driven electric DC motors in many forklift trucks permit portable power (no power cables). Golf carts use portable DC batteries too.
It depends on your application. Fans, appliances, and clocks would benefit from a/c because in these cases a/c motors are less expensive and use the a/c frequency to regulate rotational speed. If your application involves speed and or position control (tredmill, hard drive, electric vehicle) then the better motor would be DC.
AC induction motor don't require brushes and are extremely rugged. These motors operate at a RPM of 120* Frequency/ pole pairs minus the slip speed. An induction motor with one pole pair will operate just below 3600 RPM ( somewhere around 3400 RPM) at a line frequency of 60 hertz. The slip speed is usually about 150 RPM or so. It varies depended on how heavily loaded the motor is. DC motors require brushes and a commutator or some type of electronic control. This makes these motors more fragile. The speed is variable depending on load which may or may not be useful.