What contributes most to a change in how an amplifier sounds?

BTW: I love BAT equipment for preamps and power amps. Both tube and solid state. I think Victor Khomenko knows what he is talking about.

What changes are audible in a power amplifier?

Sorry to disappoint, but not anything a user can alter, or even measure. It takes intimate knowledge of the circuit (and the design philosophy) to make these changes.

First, a high-current driver section that can rapidly charge and discharge the capacitance of the output section. These are often under-designed in both transistor and tube amps. The driver current affects slew rate, cleanness of Class AB transitions, and distortion above 1 kHz.

Secondly, avoiding current fluctuations in the power supply (caused by Class AB switching transitions in the output section) from affecting the input and driver sections (which are typically in Class A). Power supply variations affect the subjective quality of dynamic impact and clarity with dense program material.

Thirdly, in feedback amplifiers, adequate phase margin, preferably well in excess of "textbook" stability minimums. This prevents reactive speaker loads from degrading and stretching out the settling time of the amplifier. Settling time, by the way, is the time required to recover from a slew-transient overload. This affects the amp-speaker interface, and why some speakers don’t "get on" with some amplifiers.

All of these aspects are audible to the amplifier designer, and appear on internal measurements of the amplifier’s functions. They do NOT appear on external measurements of a "black box" under test. Unless you have designed amplifiers yourself, you, or a reviewer, are not likely to hear what changing these parameters sounds like. If you have, though, they are pretty obvious.

P.S. Reading between the lines, Class A operation and multiple independent power supplies makes for clean and stable amplifiers that have good subjective results. But ... true thermal Class A operation greatly restricts power output, and multiple independent power supplies also raise the price.

Class D operation sidesteps the annoying Class AB artifacts, but then you get deep into designing pulse-width modulators that are unconditionally stable, resist transient upsets, have good phase margin, and also have low distortion, even under dynamic conditions. Basically all the challenges of designing a state-of-the-art ADC and DAC that can also deliver power into complex and nonlinear loads.

Everything for high end equipment. 
The best design & parts to meet some specs the cheapest way possible for mid fi equipment. 
 

An amplifier's "sound" is only apparent when connected to cables and a speaker. So it could depend on the capacitance of the speaker cable, or any steep phase angles the speaker has.

And, as mentioned, the power supply, which is why  the average receiver will not drive say, a Magnepan or some other type speaker  (hybrids, electrostatics and other exotic types) that places demands on the amp it usually can't handle.

A lower level of Dynamics, flatter line, is expected when impedance is lower?

@emergingsoul Dynamics comes from the signal not the amp. When it seems like the amp is more 'dynamic' the chances are extremely high that what you are hearing is actually just distortion masquerading as 'dynamics' due to how the distortion interacts with the human ear.

Class D operation sidesteps the annoying Class AB artifacts, but then you get deep into designing pulse-width modulators that are unconditionally stable, resist transient upsets, have good phase margin, and also have low distortion, even under dynamic conditions. Basically all the challenges of designing a state-of-the-art ADC and DAC that can also deliver power into complex and nonlinear loads.

@lynn_olson If you design a self-oscillating class D amp then you satisfy all these requirements. In a self oscillating amp you intentionally exceed the phase margin by adding so much feedback the amp goes into oscillation as soon as its powered up. The feedback loop is designed to only allow one solution for the oscillation, which is used as the switching frequency. This has the benefit of allowing much higher feedback without the problems caused by lessor amounts and having it poorly applied. It also solves the problem of noise caused when the switching frequency drifts. So this allows the amp to be dead silent even on horns.