Without knowing the answer, here is my guess: the way a flake grows depends on a number of environmental factors. Although these factors, like temperature, humidity and wind, all vary as the drop/flake falls down, the factors are nearly constant over the size of the flake. Each part of the flake experiences the same environment and as a result all parts grow in the same fashion.

My alternate guess is that snowflakes actually aren't symmetrical, although they display many local 6-way symmetries.

Edit: ah, the answer by jerf verifies that these guesses were reasonable.