I have some electronic training sets here at work that allow you to build circuits and use different voltages to power them. I tested 4.5V and 6V since these are fairly close to what we use in RC. I applied these voltages to fixed resistors, relay coils and DC motors. In every case, there was more current flowing when hooked to 6V than when hooked to 4.5V. The 6V circuits clearly used more power. The measured numbers were identical to what I calculated they should be using Ohm's law.

Of course the 6V circuits were clearly more powerful. This is good if you need the servo torque, speed, etc.