In the very general sense, the servo motor provides precise positioning control by use of a feedback element, for example encoder or resolver, the spindle motor is velocity controlled.
They both can consist of various technologies, i.e. DC brushed, AC Sinusoidal (synchronous) and Brushless DC, the spindle motor can also be a common AC induction motor type.
In some cases the spindle motor can be made to accurately position by means of feedback to the controller for functions such as orientation for tool changing and can even be done with induction motor technology, usually 3 phase type.
In the case of the servo motor, PID feedback loop allows interpolated moves between axis.