Yes and no. No two snowflakes are exactly identical, down to the precise number of water molecules, spin of electrons, isotope abundance of hydrogen and oxygen, etc. On the other hand, it is possible for two snowflakes to look exactly alike and any given snowflake probably has had a good match at some point in history. Since so many factors affect the structure of a snowflake and since a snowflake’s structure is constantly changing in response to environmental conditions, it is improbable that anyone would see two identical snowflakes.  

Why are snowflakes symmetrical?

First, not all snowflakes are the same on all sides. Uneven temperatures, presence of dirt, and other factors may cause a snowflake to be lop-sided.
Yet it is true that many snowflakes are symmetrical and intricate. This is because a snowflake’s shape reflects the internal order of the water molecules. Water molecules in the solid state, such as in ice and snow, form weak bonds (called hydrogen bonds) with one another. These ordered arrangements result in the symmetrical, hexagonal shape of the snowflake. During crystallization, the water molecules align themselves to maximize attractive forces and minimize repulsive forces. Consequently, water molecules arrange themselves in predetermined spaces and in a specific arrangement. Water molecules simply arrange themselves to fit the spaces and maintain symmetry.