Not all ice is like that - for example, if you freeze water that contains tiny water bubbles due to a faucet aerator, the resulting ice would likely look white because the tiny air bubbles get frozen too, their edges forming multitudes of tiny prisms that produce all colours of the rainbow, averaged as white or bluish white.

Snowflakes are made of very fine structures of ice, thus also forming such prisms, also looking white as the result. Because their structure ranges widely, some snowflakes are more transparent than others, but almost all look white regardless. Snow is in fact semi-transparent too, easily evidenced when a thin layer of it covers a coloured surface, and also by shining some light through it.

Under a good microscope, a small portion of a snowflake may look like a transparent ice structure, sporting rainbow colours on the edges. However as you zoom out, finer parts blur and start to look more white. Some snowflakes have more finer details and impurities that can be captured with average optical microscopes so they appear white even magnified.

'White' colour itself is just how we describe achromatic (non-tinted) light. Further, human eyes adjust to lighting colour to treat it as white. If you project tinted light onto snow, it would of course look tinted. However, it may look less tinted than paper in similar conditions because snow tends to reflect light coming from every angle better.