A clear ice cube and a pile of fresh snow are made from the same basic substance.
One can look transparent.
The other looks bright white.
The difference comes from structure.
A solid piece of ice can transmit light
In clear ice, light can travel through a relatively continuous material.
Some light reflects from the surfaces, but much of it can pass through.
That lets you see into or through the ice.
Snow contains countless ice surfaces
Snow is made from huge numbers of small ice crystals.
Between those crystals are air spaces.
Light entering snow hits surface after surface.
At each boundary between ice and air, some of the light changes direction or reflects.
Repeated scattering sends light back toward you
After bouncing and scattering through many crystals, much of the incoming visible light returns toward your eyes.
The different visible wavelengths are scattered broadly enough that they arrive together.
Your visual system interprets that combination as white.
This is another example of structure changing how light looks, much like the thin film in rainbow-colored soap bubbles changes which wavelengths reinforce one another.
Why can packed snow look different?
Pack the snow tightly and some air spaces disappear.
Compress it further and the structure becomes more like solid ice.
Now light can travel differently.
Dense ice may look clear, gray, or blue depending on thickness, bubbles, cracks, and impurities.
Why can old snow look dirty or gray?
Snow collects dust, soil, soot, plant material, and other particles.
Those materials absorb and scatter light differently.
The snow is still made of ice, but the added material changes the color you see.
Why do glaciers often look blue?
In large dense masses of ice, light can travel through enough material for red wavelengths to be absorbed more strongly.
More blue light remains.
That creates the striking blue seen in some thick glacial ice.
So snow is not filled with white pigment.
It is a giant light-scattering pile of tiny clear crystals.
And that’s the short explanation.
Related explanations: Why do clouds stay in the sky?, Why does the ocean look blue?, and Why do soap bubbles have rainbow colors?