Headroom is available clean power in excess of what you use. You need lots of headroom over, say, dialog levels when listening to movies, because the peaks may call for 100 times more power. (And dialog probably uses half a watt or less in most setups . . . even at "reference"). If I have enough clean power to hit the peaks without distortion at my listening levels, headroom above that is a lovely thing to think about, may make me sleep better at night, but cannot yield an audible difference. Look up Ohm's law. Voltage, current and resistance (called impedance in AC circuits) are subject to a strict mathematical relationship, immutable as gravity, at least on the physical scale of wires, speakers and amps (versus subatomic or interstellar size ranges) and at temperatures significantly above absolute zero. Unlike gravity, people who don't believe Ohm's law really applies are unlikely to think they can fly out of a 10th story window and crash to their deaths, or otherwise meet horrific ends as opposed to people who don't believe in gravity.
What the power amp multiplies is voltage (and the speaker responds linearly to voltage changes). Current (amps) is needed from the amp, too, but the current for any given voltage is fixed by the speaker's impedance. It's more complex than a DC circuit with a resistor, because the speaker's impedance is different at different frequencies, and since real content has constantly-changing frequency, umm, content, the net impedance the amp "sees" is constantly changing. But for the impedance the speaker presents at any given instant with real content, the amp cannot deliver the voltage called for by it's amplification factor and also deliver either more or less current. Physical laws of the universe prevent it from sending more current than the impedance of the speaker calls for at the given voltage, and if the amp cannot supply enough current, it the same physical laws mean it will not send full voltage. At least in some settings, that's called voltage "sag," and whether it's caused by the amp running out of current or the amp not being to swing the required voltage it's the basic culprit behind harmonic distortion from amps.
Assuming you are a human being, it's possible that the differences you perceive between the two devices are placebo effect - the same thing that causes some people to get over their headaches when they take a sugar pill they think is medicine. Man did I spend lots of years and more than a few dollars chasing placebo effects in electronics, but that's a story fo another day. It's an unavoidable part of being human, occurs, AIUI, largely on a subconscious level so we cannot control it, and may yield results that differ from the known ability of the less subconscious preexisting bias to affect our perceptions.
If the differences between your devices are real, the only way it can be power related is if one amp has detectable (even if you don't identify it as distortion) distortion. If there is an audible difference at lower levels, one of the devices (or, I suppose, both) is not behaving linearly. Some amps are designed not to be linear in the audible frequency ranges. For example, in the 80s, AIUI, lots of British higher-end amps rolled off the high frequencies a bit. That difference would be audible on any speaker that did not roll highs off itself so much that it masked the amp's nonlinearity. Sounded good to lots of folks, especially with early digital, but technically not accurate amplification. Of course if some EQ is turned on in the preamp section of one of the devices, the system's output will be as nonlinear as the EQ calls for and that could explain the difference. Audyssey certainly qualifies.
AIUI, another possibility, not likely with modern solid state amps, is a high enough output impedance from the power amp that it's otherwise linear frequency response varies when it's presented with a difficult load. (while AFAIK
you can just plug the impedance number into Ohm's law where resistance would go to calculated instantaneous current at any given voltage, impedance is a more complex phenomenon, containing not only a resistance element but other attributes, all of which vary with frequency, and that may make the amp respond differently at different frequencies. The folks at Stereophile.com do bench testing of amps they review. With very rare exceptions, the solid state amps, driven into what is supposed to be a very difficult test load, show frequency response variations of less than a tenth of a decibel (dB), a difference that, AIUI, is generally considered utterly inaudible. But a very few solid state amps (usually exotic high high dollar babies) and more than half of tube power amps will have variations that may reach half a dB or more, which, if the dip or peak is broad enough, may be noticeable in critical listening.