There are snowflakes, and there are snow crystals. A snow crystal is a singular crystal of ice, while "snowflake" is the all-encompassing word for either a single snow crystal, or a bunch of snow crystals stuck together, "or large agglomerations of snow crystals that form 'puff-balls' that float down from the clouds," according to Kenneth Libbrecht, a physics professor at the California Institute of Technology, who specializes in snow crystals. Just as a blue jay is a type of bird or a Corvette is a type of car, a snow crystal is a type of snowflake.
So, could two snow crystals be alike? It's complicated. Basically, no. As Libbrecht explains in his oft-cited blog post about the subject, most small snow crystals contain 1018 water molecules, and some of these water molecules are slightly abnormal. "These unusual molecules will be randomly scattered throughout the snow crystal, giving it a unique design," Libbrecht writes. "The probability that two snow crystals would have exactly the same layout of these molecules is very, very, very small. Even with 1024 crystals per year, the odds of it happening within the lifetime of the Universe is indistinguishable from zero." By the way, that 1024? That's one septillion, or a trillion trillion.
There is a potential exception, Libbrecht says: "a snow crystal with only a handful of molecules," like only ten. That's the nano-snowflake, and considering most snowflakes have 1018 molecules, it's not common.