The "distortion" the LM386 is producing is "clipped" distortion, and probably looks something like this if the output were viewed:
That is a "hard" clipping at the tops and bottoms, and we perceive that as a "metal" type of tone.
When you overdrive a vacuum-tube amplifier, it also distorts the waveform, but the edges are much more rounded, and thus, pleasant to listen to.
Here is some more information on musical distortion, including a blurb about the legendary Ibanez TS9 "tube screamer."
The bottom line is, hard clipping will always sound "metal", and soft clipping will sound more pleasant. It is possible to achieve soft-clipping with solid-state electronics, but it's a little more involved than simply overdriving an input. DIYstomboxes have many schematics available for various types of distortions and other effects.