Distortion is the mother of effects in modern rock, metal, and many other styles such as punk, grunge, or power pop. Distortion is nothing more than the alteration of a clean signal. This distortion works differently in tube and transistor amps, affecting different types of harmonics and producing a radically different sound.
A soft distortion is good to get good dynamics (more accentuation) so virtuous guitarists prefer moderate levels of distortion so their touch is not affected and so the nuances more distinguishable. In contrast, stronger distortion can help gain bass presence (perfect for metal rhythms) or intentionally make the sound "dirtier", exaggerate the highs, and so on.
What is the difference between distortion and overdrive?
The difference resides mainly on the enhancement and alteration of the harmonics. The sound of the distortion is "thicker" but it still sounds very defined and fairly clean. An overdrive is a soft effect, a distortion of the wave that affects the second harmonic but not the third, so that the wave it is not so heavily altered.