Snow forms high up in the atmosphere, where the temperature is low enough for water to supercool, meaning it remains in a liquid state even below the normal freezing point of water. When a few molecules in a supercooled droplet of water come together in a latticelike arrangement, this forms a nucleus about which the droplet freezes, creating a snow crystal or snowflake. Occasionally a particle of dust or other tiny solid in the atmosphere can function as the nucleus around which a snowflake forms.
The frozen crystal then grows by collecting water from the surrounding air. It may collide with and form clusters with other such crystals, making a larger snowflake.
Eventually the snowflake gets heavy enough to fall toward the earth. As it falls, it passes through different altitudes of different temperature and different humidity. It may partially melt and refreeze multiple times. This process determines the shape it takes by the time it lands on the ground.
The snowflakes might come out as flat planar crystals, as three dimensional dendritic crystals, as long thin pencil or needle shapes, or as long and thin like that but capped with planar crystals at each end.
Actually the most common form that snowflakes take after being formed and reformed and altered so much as they fall is that of irregular shapes that lose their symmetry and elaborate design. So the kind of snowflake we see on Christmas decorations and such is very much the exception rather than the rule.
Even once on the ground, the snow continues to change form through a process called metamorphism. The individual snowflakes gradually lose their shape and become rounded off, and then meld with the other snowflakes in an undifferentiated mass.
Due to the irregular manner in which snowflakes form and change as they fall, and due to the number of molecules they contain (typically about 1,000,000,000,000,000,000, which is 1 quintillion), there is no realistic likelihood that two snowflakes will ever be precisely identical. Hence that bit of common folk wisdom turns out to be (probably) true.
Not only can snowflakes differ almost infinitely in their final shape, they also can differ a great deal in size. The largest snowflake ever recorded (at Fort Keogh, Montana, in 1887) reportedly had a diameter of 15 inches.
Sources:
Donna Thacker, “How Does Snow Form?” eHow.
“How is Snow Formed?” The Gemini Geek.
“How is Snow Formed?” Let Us Find Out.
