Nice question … I think it’s because the snow forms small separate crystals. Each of these crystals has surfaces which light can bounce off. So, when light shines on snow it is bounces off these surfaces in many different directions, and it is this diffuse bouncing (scattering) which gives it the white colour. Note that it isn’t like a mirror, because the bouncing is in random directions – a mirror bounces light in consistent directions (you can work out how by reading about Snell’s Law).
Ice has a more regular structure, which allows light to pass through without being scattered. It doesn’t have so many surfaces to scatter off, unless lots of bubbles are trapped in the ice.
Water ice is really interesting – one of the few solid that is less dense that its liquid (which is why icebergs float – this wouldn’t happen in seas made from other molecules). Also, while we’re familiar with one form of water ice, there are actually many other ways that the water can solidify, particularly under high pressures. They have exciting names, like Ice IX, Ice X, Ice XI … 😉