The power the mill consumes is a function of the ore and/or ball charge geometry and the mill speed.  The motor only supplies the amount of power (or torque really) that the mill demands, and we add some extra power to the motor to deal with 'upset' conditions like a change in ore density or a surge in ore level inside the mill.  Detailed explanation of how motors are chosen is available from here: 

https://sagmilling.com/articles?topic=drives

The "pinion power" is the mechanical power available at the output shaft of the motor and is related to the torque required to turn the mill and the mill speed.  If you remember back to your electrical engineering courses, the power available in a circuit drops due to losses as you move down the circuit (because each electrical item causes a voltage drop, or a resistance).  The input power you measure to the motor will be higher than the mechanical 'pinion power' due to the electrical losses in the motor.  And if you measure the electrical power farther upstream, you'll get an even higher power due to the accumulated losses of downstream drives, conductors, transformers, and so on.

In addition to Alex's comments, there are additional "losses" in the mechanical side.  The input to the gears  and the output from the pinion to the shell are always different, with mechanical losses of up to 10% (higher with poor maintenance). 

Then there are losses due to rolling friction on the trunnions, etc.  Then comes the issue of standard motor sizes.