How to increase the power of an electric motor?
Same top speed (RPM).

Those induction motors have speed dependent on the incoming mains frequency, so they will be nominally the same, regardless of rated power

Same Voltage.

They are intended to run from the same voltage.

Difference in current required. What is physically different between the two? What causes one to draw more current than the other? Is it simply more winds on the coil? Thicker gauge wire?

Probably a different number of turns on the stator, which might require a different guage wire.

Larger stator/rotor?

Any given frame size motor will likely use the same diameter rotor. You have a frame 80 and frame 90 motor there, so they will be slightly different.

How is it they have the same max RPM?

Their max rpm, under no load, will approach 1500 rpm, this is the synchronous speed with a 50 Hz mains frequency.

Shouldn't more power mean higher speed in the absence of other variables (e.g. weight)? What is the extra current spent on if not speed? Torque? Or is the extra current only drawn under load; in response to a physical resistance (i.e. drag) on the driveshaft?

Speed is nominally constant, due to the AC frequency being constant, so the other variable, torque, changes. 'Slip' is defined as the % reduction in speed below synchronous. The current drawn by the motor is more or less proportional to the slip. As the motor load increases, the speed will drop, the slip increases, the current and so torque increases, until it reaches the rated speed of 1440 rpm at the rated power, either 1 or 2 hp depending on the motor.