Snow appears as white due to the way light behaves when it hits the snow. Snow is actually made up of lots of tiny ice crystals all stuck together. When light hits an ice crystal, it goes through it, but not directly -- instead it comes out at a different angle than that at which it entered. When you get a bunch of ice crystals all stuck together (snow) the light hits the first ice crystal, is bent slightly and sent into another ice crystal and so on. The end result is that the light bounces right back out of the snow. The same thing happens to all different light frequencies hitting the snow, so basically, all possible colors are "bounced back" off the snow. The combination of all these light frequencies is white (the color of all colors in the light spectrum combined) and thus the snow appears to be white.
So...when the snow melts, and it is no longer ice crystals, you no longer get the same "bouncing back" effect from the light and -- no more white.