Class A and Class A/B are terms that describe how the power tubes work within the power section of a guitar amplifier. To properly explain the technical differences between these classes of operation would require a lengthy discussion. However, in short, the differences can be summarized as follows:
Class A/B amps tend to have greater dynamics, sound punchier, tighter, cleaner, and have cooler running tubes. The Class A amp sounds more vintage and squishy, because it’s compressing and distorting more. Tubes in a Class A amp tend to run hotter, as well. For the same given tube compliment, Class A/B will produce two-to-three times as much power as Class A. An example would be an amplifier with two 6L6s in the power section. Operating in Class A, the maximum power we could expect would be around 20 watts, while operating in Class A/B would easily yield 50 watts.
Just for tube life alone, I believe Class A/B is the way to design any amp. The amp will run more efficiently with more power and you’ll enjoy not having to replace power tubes as often. If the tonal characteristics of a Class A are desired, an A/B amp can be carefully designed to do that (the Soldano Astroverb is a good example of such a design).