Keep in mind, I'm going to attempt to state this in the most neutral way possible.
In my experience, with normal dynamic headphones (like DT880, AKG's, ect ect)) more power does not make a damn bit of difference. Most dynamics don't ever reach power requirements high enough to warrant a heavy-duty amp, so you'll end up just buying an amp for its own "sound" (distortion.) Kind of like buying a tube amp or warm solid state to balance out the HD800's coldness...stuff like that. Again, this is my experience, but I found planar headphones react to more power in way that increases how well the bass sounds and feels as well as the soundstage size. A great example is my HE400s. The bass sounds like it is missing something out of both the O2 and the Magni (and they do sound the same to me,) but sound much much better out of the Asgard 2. However, besides the coloration other amps provide, I didn't notice an improvement when I moved onto a lot more expensive and more powerful amps. However, something like the Alpha Dog and LCD 2 sounded a tad souless out of the Magni/O2 and only a bit better out of the Asgard 2 (but not great.) Once I supplied more power, ala Lyr 2 and the Burson Soloist, I felt the bass a lot better than I did before. Seeing as there is no proof besides the numerous accounts of people hearing the difference, it is all just subjective (though some people also state that planars need 150% more power than math would dictate...you can thank Audiobot for that.) YMMV. Headroom's definition seems to bounce from person to person, but I consider it "enough power to drive the headphones close to or at their dB maximum." Allows you enough room to say goodbye to your hearing while maintaining a nice control over your volume. Not to mention the benefits I mentioned for the planars.
Another example of "not enough power to a planar" is my Mad Dogs. They sound a bit lean in the bass on my Cayin C5 on high gain and bass boost, but sound nice and full on my Burson. But, as I said, YMMV.
Objectively, however, most people would say "if there's enough in the specs to get it to normal hearing range, it is driven sufficiently." However, that is both true yet a farce. If you try to drive the headphone to its maximum volume, but it starts distorting, than the amp doesn't have enough power to keep it from clipping (I.E., current is not enough.) That means it you're a lot closer to the amp's limit in power when you listen to it at normal levels than if you were listening to them through a more powerful amp. Because you're a lot closer, there's more noise and distortion. Despite it being incredibly subtle (and I think the THD is just below a percent for the Alpha Primes on the O2,) the distortion can sound wrong. Granted, a person shouldn't be able to tell something like this, but apparently humans can't see more than 60 FPS yet 144hz is a noticeable improvement over 60hz...so who knows what the brain picks up that we can't out-right tell is there (kind of like how it is arguably noticeable going from a 94/24 to a 192/24 DAC even though it shouldn't make a difference to our ears.)
The irony in the objective argument is that "loud enough" is the basis for what to drive headphones do, despite that being a subjective measuring stick. Even though one person may think the normal listening level (85 dB,) another may think that is too low.
Either way, outside what could or could not be true about more power with planars, an amp's color can help or hinder a headphone's sound.
