I intended to use the word flat instead. Yes, I know that they have a thickness and therefore are not two dimensional but why do snowflakes not grow in a three dimensional lattice type structure?
The pictures/photos you see are ideal ones. Nature tends to be rather haphazard. That said I guess the question as to why it has a tendency to spread out in one plane is fair enough. Perhaps once it starts it's more difficult to grow perpendicular to what has already formed; easier to keep going the way it already is ?
What atmospheric conditions prevail to produce, on one occasion, a flattish snowflake and, on another, spherical crystal s called hail. What compels the water molecule to adopt these two different configurations.