When it comes to guitar amps there are no absolutes, but a few things can be worked through.
Class B means, in the context of a guitar amplifier, a design that has a symmetric “push pull” output stage. This requires a different design for the output transformer as well, its input (primary) winding is two winding joined in series (aka a centre tapped winding). When the signal goes positive, one tube conducts current through one side of the output transformer’s primary winding, when the signal goes negative the other tube conducts current through the other side of the output transformer primary.
A Class A guitar amplifier has a single tube that is conducting half of its full forward current when there is zero signal. When the signal goes positive it conducts more, when the signal goes negative it conducts less. Again it controls the current passing through the output transformer primary. The downside is that even at idle the tube is conducting half full current, and thus runs much hotter than a class B.
Either way the output transformer is driven with varying current in the primary which is reflected in the output at a low enough impedance that is can drive the speaker.
This is all there is from a basic point of view. However when it comes to guitar amps we are usually looking to overdrive the output stage. At this point the difference between a symmetric and asymmetric output stages becomes important. To a good first approximation the symmetric stage will clip the waveform in a similar manner on the positive waveform and the negative. This is because both the positive and negative sides drive the tube to higher conduction as the signal rises in amplitude, so the final waveform is the combination of mirror images. A Class A output stage does not clip symmetrically. The transfer function of a tube looks quite different at the two extremes - driven fully versus cutting off. So the output waveform when driven hard is not symmetric.
The mathematics here contains the core of the sound. You can always decompose a waveform into two parts, the half wave symmetric and the half wave anti-symmetric parts. The Class B amplifier has an output that has very low half wave anti-symmetric components, whereas the Class A amplifier output has a mix of both.
Half wave symmetric waveforms contain no even harmonics. Half wave anti-symetric waveforms contain both odd and even harmonics.
The bottom line is that an overdriven class A amplifier will have an output with significant even harmonics, and an overdrive Class B will be predominantly odd order. Odd order harmonics generally sound harsher, as the higher order ones clash harmonically. Low order odd harmonics are however good. After all the third harmonic is the fifth up an octave. Even harmonics are more rounded in sound. The second harmonic is the octave, the fourth up two octaves, and the sixth is the fifth up two octaves.
But the preamp and general gain layout of the amp is going to dominate the sensitivity of the amp. Also there are other aspects of the output stage that govern its sound and response. The use of negative feedback is critical. There are huge difference here between different amp designs. So it is essentially impossible to generalise about the difference between A and B designs based only on that one aspect of the design.