Connect motor control board's 4 terminals to brushed motor
M+ and M- are the only ones that go to the motor tabs. Either way is fine; Just changes assigned direction of rotation. Doesn't look like anything else goes to the motor.
That PCB on the motor doesn't look like it does very much. Just noise supression measures it looks like. That wire on the motor can probably be ignored if you want. It looks like it is just a midpoint connection between the noise filtering caps so you could connect it to the case or chassis if you want.