It all has to do with air. Objects get their color by absorbing or reflecting different wavelengths of light. For instance, a banana absorbs all wavelengths of light except yellow, which it reflects back to your eye. White is the result of reflection of all colors.
Snow and snowflakes are ice crystals mixed with air molecules. The air gaps between crystals, along with the crystals' complex shapes, bounces the light beams around so much that all the wavelengths eventually get reflected out, giving it a white color.
Meanwhile, ice has very little air between the frozen water, allowing the light to pass through with minimal reflection -- although thick ice will eventually absorb the red colors, giving it a blue tinge.
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