symmetrical-ski switch better, some skis have symmetrical flex with shape, helps with tricks in both directions
asymmetrical- ski forwards better, carve better, most have different flex patterns, handles all-mountain terrain
creativity comes to life in the darkness-surreal and dj balance
And lower swing weight, usually, because of the dead center mount.
I still maintain that the difference between near symmetrical skis and true symmetrical skis is barely noticeable if the other specs of the ski are held constant. I'd even say that near symmetrical skis are a bit better just due to the fact that I'm not usually riding switch 50% of the time on snow, and also consider the fact that your center of gravity and way you initiate turns is completely different when going switch vs forwards (and enter here the huge argument as to whether the symmetrical ski helps or hurts this, another time).