Class A into Class AB

It's important because most loudspeakers are only drawing 1 to 5 watts most of the time which for many amps keeps them in the class A operating range.

According to Douglas Self there is "optimal bias" where distortions are the lowest (he shows charts). Increasing bias above it (overbias) increases distortions.  It is trading of crossover distortion with "gm doubling" distortions.  Transconductance (gm = voltage to current gain) suddenly changes when two output transistors conduct (class A) instead of one (class B).  Perhaps transconductance change is slightly more audible when level is a little higher.

Also, anything can be fixed with deep negative feedback.  It lowers THD, IMD, lowers output impedance and widens bandwidth.  Deep NFB also produces TIM (Transient Intermodulation) distortions  - an overshoot in time domain, that produces higher order odd harmonics making sound bright and unpleasant.  
Class AB amplifier has voltage gain of few thousand without feedback while class A amplifier has gain of few hundred.  Increasing bias with amp that already has higher gain won't help much - NFB damage (TIM) is already done - it would have to be designed with shallower NFB - but this can make gm doubling distortions more audible.

Question for atmosphere:  Do you think amps that are only spec'd over their class A operating range, but transition to class AB outside of that range, are properly called "class A" amps?  I'm thinking of Luxman L590axii, spec'd to 30W at 8 ohm, 60W at 4 ohm, and class A over that entire specified range.  But nevertheless capable of going to about 90W, 8 ohm outside of class A in AB, per at least two magazine measurements (which I forget now, maybe Stereophile and HiFi News). 

Would you call that a class A amp or a class AB amp?

The difference in how A achieves zero volts, and how B does it.

With Class A, 0 V is achieved by the + and - transistor banks being fully on.  Because they are balanced, there's no voltage, but full current and a whole lot of heat.

B achieves 0V by both banks being off.  No current, no heat. Trouble is that the first couple of volts of a transistor are not linear.  They need to reach a minimum threshold to turn on and respond to the input.

Class AB is a compromise where 0V is achieved by both banks being on, but only a little, and when voltage swings enough one bank turns fully off.  High efficiency, low distortion. 

Mostly it is marketing lingo, for a biased amp.