Actually, while it is true that power is a function of voltage and current (W = V*A), the actual variables that affect power output are the load (impedance curve of the headphone) and the current that the amp can provide at that load (assuming, for a certain volume setting, that the output voltage is constant). It isnt really possible for 2 amps to deliver the same power output and have different current levels, unless the impedance of the headphone is changing from one amp to the other - if 2 amps are providing different current, they are providing different output power.
A powerful amp does make a difference b/c the impedance of even sensitive headphones may not be linear - nor may it be fully resistive in nature. As such, the ability of an amp to handle varying loads does affect the output sound. Which is why a relatively inexpensive, low-current, high-global-feedback op-amp based unit is not going to have the same control over a difficult load (ie, varying impedance, not low sensitivity) as a beefier amp with more current capabilities. In other words, it isnt about the amount of current - it is about ability to handle changes in current requirements due to varying loads - especially once phase issues caused by L/C components of the load enter the picture.
