It is the volume of copper that makes the difference. Volume is limited by the space between the poles. Bigger motors can be more powerful.
If you double the cross-section of the wire, you will only get half as many turns in the available space, but you will be able to run twice the current. The shaft torque of the motor is proportional to the strength of the magnetic fields. That is, amperes*turns.
Power is energy flow, measured in watts = joules / second.
For a motor, power is proportional to the shaft torque * RPM.
So if a motor spins at twice the speed, with the same torque, it will produce twice the power. It will need twice the voltage to overcome the internal back EMF produced by the motor at twice the speed. Higher voltage means thicker insulation, which reduces the section available for copper.