To see why snowflakes look like they do, consider the life history of a single snow crystal: the story begins up in a cloud, when a minute cloud droplet first freezes into a tiny particle of ice. As water vapor starts condensing on its surface, the ice particle quickly develops facets, thus becoming a small hexagonal prism. For a while it keeps this simple faceted shape as it grows.
As the crystal becomes larger, however, branches begin to sprout from the six corners of the hexagon. Since the atmospheric conditions (e.g. temperature and humidity) are nearly constant across the small crystal, the six budding arms all grow out at roughly the same rate.
While it grows, the crystal is blown to and fro inside the clouds, so the temperature it sees changes randomly with time. But the crystal growth depends strongly on temperature, thus the six arms of the snow crystal each change their growth with time. And because all six arms see the same conditions at the same times, they all grow about the same way. The end result is a complex, branched structure that is also six-fold symmetric...