This question comes up all the time. There were many good answers but lets get real. Technically it is cheaper to run on 240 volt. This has to do with wiring resistance. For motor operation resistance is just s waste of energy in the form of heat. 240 volts require half the current that 120 volt require. So less heat is generated which means it is more efficient. there are many other factors to efficiencies like wire sizing, material, and windings. Unless you are a factory with lots of motors, don't worry about it.
So what does 220 volts buy you? Maybe nothing unless you have a large load. First let me say I as a rule of thumb of 7.5 amps per horse power (HP) at 120 volts. This is a good number for a good motor with its inefficiencies. Lets use my table saw as an example. It has a 2 HP motor. Under normal load, that means it will draw 15 amps (W=I*E). It is normal to exceed that 15 amps calculation when you load the motor.
When I ran oak through the saw it was not unusual to trip the 20 amp breaker. 20 amps - 15 amps only leave a 5 amp head room. If the breaker didn't trip, the thermal over load on the motor did. Now what about 220 volts. If you double the voltage the current is cut in half. So when I convert my table saw to 220 volts, the current required is 7.5 amps instead of 15. Now with the same 20 amp breaker, I have 12.5 amps to spare before tripping. After changing to 220 volts on my table saw, I have never tripped the circuit breaker or thermal over load.
My feeling is this, if your motor is over 1 HP switch to 220 volt. You won't save any real money but your motor will be happier.
Sorry for the long winded response. Like anything else there is a lot more to this but for the home hobbiest, it doesn't amount to a hill of beans.