Why are snowflakes shaped as they are?
Now, as I understand it, all processes proceed so as to maximize the "randomness" of its constituent particles. (Oversimplified version of the Second Law of Thermodynamics, yes, I know... just don't chew me out in the comments section...)


Yes. True.


  This Law can easily be observed in, and verified by, natural processes.


Sure. Still with you.


  Now the formation of snow is a natural process, agreed? The way my brain sees it, is that water droplets ought to freeze into random, and by virtue of its "randomness", highly unsymmetrical shapes. But this is not the case here!


Nope. Let's consider a counter argument: chemical bond formation between atoms is a natural process, and natural processes seek to maximise entropy. Thus, all molecules should take on highly unsymmetrical shapes, shouldn't they?  The answer is obviously no. Same applies for snow flakes. First of all, "order" and "symmetry" aren't precluded by thermodynamics. Molecular and crystal structure emerges within the realms of laws of thermodynamics and quantum mechanics.

Like I said in the comments, the precise discussion of how a snow flake's shape merges will be quite involved. There will be a lot of variables that affect the growth of snow crystals: temperature, vapour pressure humidity, wind speed and direction etcetera. 

The final shape of the snow flake would depend on the precise path it followed, and the conditions it experienced as it descended, and is impossible to predict/calculate.

Of course, the underlying six-fold symmetry comes from the arrangement of water molecules in the ice crystal lattice. 

Also, there are no natural processes coordinating the growth of the arms to be perfectly symmetrical; their growth is governed by simple statistics. 

The arms of a snow crystal all grow independently, and grow under randomly fluctuating conditions; however, the effects of these fluctuations are somewhat averaged over the entire snow flake as it tumbles through the sky, and you end up with a somewhat regular shape.

There is in fact, asymmetry in snow flakes. Just take a close, hard look at a photograph

 

Sure there is some symmetry: the snow flake has 6-major arms, and then each arm has sub-branches, and each sub-branch branches out further. The overall branching pattern is the same, because of random averaging--the snow flake randomly rotates and reorients as it falls, each arm feels (on average) the same conditions. But are each of the sub-branches or the sub-sub-branches identical in shape, number, size etcetera? No.