A better comparison would be actually measuring the volume to be equal and switching back and forth between the amps.
Well the next questions would be why do they sound different (or sound at all) and how (technically).
Of course, though I don't have the equipment to do that. I wish I did, it would make doing proper comparisons much easier.
The two amps I was testing were the Matrix M-Stage and the Emotiva XPA-200. The XPA-200 has more bass, deeper bass (perhaps a misinterpretation from or a result of the increased bass response), and highs that I can listen to for longer at louder volumes (less emphasized). I noticed the M-Stage had treble that was slightly uncomfortable to listen to after moderate periods of time, and my first impulse was to turn down the volume. This is interesting because if you ask most people, the M-Stage is the one with reduced highs and increased bass response, while the Emotiva is a "bright" amp. I frequently listened to a song I use to test equipment, The Bridgeport Run by The Flashbulb. It's excellent for testing specific aspects of a headphone such as transient response and detail. I noticed a more exaggerated (whether it was technically accurate I can't say, since it's electronic music) dynamic range as well as faster transient response (leading to the "punchy" type sound). I also was able to better distinguish the individual noises in the song, it's a very difficult song for most headphones to reproduce because it's just a ton of varied noises on top of each other with wild swings in volume and duration, which usually end up blending together in gear that doesn't have the ability to properly reproduce it. Essentially, it sounded "clearer", and the noises did not blend together as much with the XPA-200.
As for why, because I don't have the equipment to do proper measurements and official measurements can be scarce and often incorrect, I don't think I have enough information to say. The only thing I am willing to say is that there is a possibility that during dynamic peaks, I am using enough power to noticeably increase distortion on the M-Stage during those peaks. This is often incorrectly interpreted as sounding "loud" and could explain why I am able to comfortably listen to the XPA-200 at a higher volume, why the M-Stage is difficult to listen to at higher volumes, and why my impulse is to lower the volume on the M-Stage and not on the XPA-200.
And are the same differences in sound heard with easy to drive headphones? Meaning it's likely a characteristic sound of the amps themselves, and not necessarily because one is driven by the speaker amp, and the other isn't.
Unfortunately, the only other headphones I have to test are the Denon D7000, which are 25 ohm and 108db/mW and I'm not willing to power them with a 150w speaker amp, sorry.
I will say, though, that I love the way the D7000's sound on the M-Stage.