Symmetrical clipping sounds more like solid state amps, while assymetrical sounds more like tubes.
Symmetrical clipping is less natural sounding, whereas assymetrical clipping is more natural sounding.
Both have their uses in different places.
Because symmetrical clipping is symmetrical, its better for overdubbing & panning, because assymetrical clipping will produce an odd set of harmonics that might end up fighting in the mix.
But when it comes to lead tones, i think assymetrical clipping wins, as it sounds richer & more warm sounding.
I prefer to use assymetrical clipping for the kind of music that i play, but for thrash & death metal, definitely symmetrical.
Anyways, two different things that has their own unique uses.