Every snowflake is different. But while each snowflake itself may be different from the one that came before it, the crystalline shapes that make up all these special little snowflakes are actually remarkably uniform.
Andy Brunning over at Compound Chem turned the data from a recent paper in Atmospheric Research into this chart, detailing the eight different kinds of snowflake structures... as well as their various 39 subdivisions.
Although we know how to characterize these different kinds of snowflake shapes and something about how they form (for instance, the more complex structures are generally seen at higher humidities and the surrounding temperature also plays a key role), what we don't know is why an individual snowflake takes the precise shape it does and not the shape of any one of its million or so relatives falling at the same time.
Don't have any snow of your own to practice on, or not eager to wade out into the blizzard with a chart and a microscope? Then test out the identification process on some of these up-close and personal looks at the crystalline structure of the snowflake.