The ECT PWR button does not necessarily make your car accelerate faster (i.e., there's no increase in power or acceleration), but it does change the logic of the automatic transmission to make it more "sensitive" the throttle position.
You can try this: In ECT "Regular mode", cruise along at some speed and then push the gas pedal down some distance (not fully). The car may or may not downshift to accelerate. Next, activate the ECT "Power" mode and cruise along as before and depress the gas the same distance as before. The trans will probably downshift immediately to bring your speed up quicker. This has nothing to do with the engine output, only gear selection.
ECT PWR will make your car feel more "lively", whereas "Normal mode" may be more comfortable to drive since the car will not change gears as readily.