Class A amps re-create the input signal more precisely than the latter, thus many say they sound better. They do however draw more current, create more heat and require large powerful tubes to do this.
A class A/B amplifer will be more efficient, draw less current and run cooler (some say at the expense of sound quality), but can provide much higher output levels with less powerful tubes.

Class A amplifers are generally more expensive, i.e. consider the VOX AC30.

Most amplifiers will fall into the class A/B category. The main difference is the way the output tubes are configured. Class A amps have an output tube(valve) that is conducting 100 percent of the time, even when an input signal is not present. Class A/B amps generally have a pair of output tubes that operate as a push pull pair. One tube conducts during positive half cycles (roughly) and the other only conducts on the negative half cycles of the input signal. It really is a lot more complex than this, but this is the general idea.
With todays technology, you'd truly have to be an audiophile to "hear" the difference between a true class A amp and a good class A/B amp.