Saturation is a defining factor in ‘that analogue sound’. It can make sounds fat and beefy, smooth and warm, or crunchy and edgy. Placed strategically on single tracks and buses, it can produce a more cohesive mix.
The most obvious effect is the addition of harmonic content, most easily heard on sounds that don’t have much to begin with, such as subby sine basses. By adding upper harmonics, a deep bass that vanishes on smaller speaker systems magically becomes audible. Generally, transistor distortion adds sharpness and crunch, and valve and tape are associated with warmth.