A motor driver uses a larger chip or discrete FETS which are able to handle larger amounts of current and higher voltages than the standard 5V/3.3V from a microcontroller pin. They allow you to control a much larger load, from a small signal.

A motor driver simply handles the power to drive the motors, whereas the logic and digital control has to be done by an external microcontroller or microprocessor, whereas a motor controller has all of the logic circuitry built in and can be controlled by a higher-level interface such as a PWM signal, USB, analogue input etc.