Janko Gravner at UC Davis and David Griffeath at the University of Wisconsin-Madison have modeled snowflake growth, as reported on this web page:
the researchers were able to recreate a wide range of natural snowflake shapes. Rather than trying to model every water molecule, it divides the space into three-dimensional chunks one micrometer across. The program takes about 24 hours to produce one "snowfake" on a modern desktop computer.
Paper, code, and movies on this modeling are available here.
Here is a 1min17sec YouTube video of growth simulations following this model;
and here is a 26sec, more colorful set of simulations.
Added. Some added detail from the G-G papers:
The building blocks for snowflakes are hexagonally arranged molecules of natural ice (Ih). Just how the elaborate designs emerge as water vapor freezes is still poorly understood.... The solidification process involves complex physical chemistry of diffusion limited aggregation and attachment kinetics....Our basic set-up features solidification Cellular Automata on the triangular lattice $\mathbb{T}$ (to reflect the arrangement of water molecules in ice crystals).
To echo Igor, it's "not so simple"!
A more physically based, 3D model is explored in the paper
"Monte Carlo Simulation of the Formation of Snowflakes," by Maruyman and Fujiyoshi
Journal of the Atmospheric Sciences, 2005.
Comparisons of the shape to "observed snowflakes" are made:
