Both breakout boards and motion controllers come in a wide range of features and pricing.
Breakout boards basically just pass signals from the software to the motors and switches. Basic breakout boards are just screw terminals connected to a printer cable. More advanced ones may have spindle speed controls and relays.
Motion controllers receive low level signals from the control software, and generate step and direction signals for the motors. They also control time critical functions like homing.
Some motion controllers actually need to use breakout boards to interface with your electronics and hardware.
Motion controllers usually are designed to work with a specific control software.
For Mach4, you can only use motion controllers that have a Mach4 plugin to allow Mach4 to communicate with them.