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.
It's more a function of the output impedance of the amp, not its horsepower as it were, that determines its ability to handle the varying input impedance of the monitor. My low impedance Merlins aren't suddenly better with a high impedance amp, just because it's powerful (ie they work better with a Clip+ than a HiFiMan 602). That's what allows an amp to handle varying loads. A sufficiently low output impedance.