Because snow is just a bunch of tiny crystals that scatter all wavelengths in all directions. When you think ice, you're thinking of a solid chunk of frozen water which refracts light the same way, albeit with a lower index of refraction. If the ice happens to be cloudy because of air or impurities, it will scatter more light and appear cloudy, all the way to the point of basically becoming snow white the more impure it is.

Edit: to explain scattering in an ELI5 friendly way, ever played with marbles as a kid? Most of them used to be transparent like glass, but we used to grab a subset and rub them against a rough surface like a pavement to make them opaque. In doing so, they would take a certain color. That's basically how light scattering works, the rough uneven surface doing the work.