As matt_black said, it's about the size.
If you take a tall, thick glass tube, it'll be transparent, right? It'll transmit the light falling on it. But make it thinner and thinner(think human hair) then it'll appear white, as it'll refract the light passing through it.
As the light from the sun(which is a combination of all colors) is white, snow appears white.
Many people think that old people's hair are white, but they're not. They get their color(black, brown, yellow, red, whatever) from a pigment called melanin, and in white hair this pigment is absent, leaving only a transparent rod of hair. It's so small it appears white.
About the ice cube part, if you've observed, sometimes there's a ice cube that's transparent on the outside, but in the middle there's a white core. That happens because it has shattered in the middle, probably because of anomalous expansion of water.