Class A is a tool in the designer's box which allows for lower distortion.
With all amplifiers, distortion is essentially what causes amps to sound different from each other, moreso than actual frequency response.
The reason for this is that the ear converts distortion into tonality. For example the 2nd harmonic often associated with tubes adds to richness; the 7th harmonic creates a metallic quality. We've know this fact about harmonic distortion since the 1930s.
Now it is the higher ordered harmonics that contribute to harshness and brightness. This in a nutshell is why solid state amps tend to sound harsher and brighter than tubes (keeping in mind that the tube amp may well have bandwidth equal to that of the solid state amp).
Class A in a push-pull amplifier allows for even ordered harmonic cancellation throughout the entire waveform right up to full power. This means it will be lower distortion- and in part due to the fact that the output devices never turn off (so no crossover artifact either). Now its often said that its that first watt that counts the most in most audio situations and this is true. When you operate class A, your distortion at low power levels is lower, and this translates to greater low level detail; distortion is not obscuring that information.
Now using loop negative feedback is good for reducing distortion also, but IMO/IME its a poor means of reducing feedback. This is because feedback, while suppressing distortion, actually introduces some of its own and its almost entirely higher ordered harmonics (plus IMD introduced at the feedback node). So any amplifier employing feedback will be brighter and harsher than real life. This is one reason why class A is important, because it is a tool that can allow the designer to make a low distortion amplifier that does not employ feedback.