Why tube amps have a special sound

Tube (valve) amps have a different tonal quality from solid state amps, which is preferred by many players. The classic tube sound is produced when the tube is overdriven and distorts the waveform. Tube distortion is characterised by 'soft clipping', as well as other effects (see Merlin Blencowe's book 'Designing Tube Preamps for Guitar and Bass').  It is difficult to emulate these effects using solid state electronics.

The upper trace shows soft clipping. The lower trace is the undistorted signal.