No two are the same. Their formation is determined by the temperature at which they form. There are some various basic shapes like radial arms, plates, and even cylindrical shafts. They take up the familiar six sided shape because of the 120 degree bond found in a water molecule and its interaction with crystalline structure.
A snowflake observed with a scanning electron microscope reveals the intense symmetry and extremely detailed crystalline structure of a typical snowflake.
For more regarding these intricate little ice plates, National Geographic has a feature with some highly detailed photos, and more regarding the relationship between temperature and physical structure of various snowflakes.