Before a snowflake ever lands in your yard, it really goes through quite an adventure, and that helps justify the saying that "no two snowflakes look alike."

To get snow, you actually need minute particles of dirt or clay in the air - typically, there might be dozens of these in every breath you take. Some of these particles have a crystal structure that sort of resembles that of ice. Invisible water vapor molecules in the air find that structure appealing, and at temperatures below freezing, some of the water vapor freezes onto those particles. This starts the process of making a snow crystal. From there, the exact details of how the crystal grows depend on the temperature and humidity of the air.

As these snow crystals fall, they can get roughed up. They might collide with each other and break up into tinier pieces. Or if temperatures are near freezing, the crystals may be coated with a thin film of water which acts a little like glue, so they'll stick together when they collide, creating bundles of snow crystals that we call snowflakes.

Given the crystals complicated lifetime, the chances that any two will form in exactly the same way are very very small. Two snowflakes may "look" alike in a general sense, but under a microscope, the crystals, like human fingerprints, will not be identical.