I think class A, just means that the amp uses a single circuit and is simpler than class A/B, it's something to do with the way the amp and the valves are actually wired inside. You can still have class A amps with more than one valve in it, aka the Vox AC30, so it's to do with amplifier electronics, which I know very little about because I am a musician, not an amp builder.
From a musical point of view class A amps sound more natrually sweeter and compressed, perfect for recording, and perfect for blues and jazz. The A/B sound is more open and powerfull in tone, and it sounds bigger, perfect for live gigs, metal and rock music.