There's an amazing amount of different types of snowflakes, as this recent infographic from the Compound Interest chemistry blog shows:

The graphic shows 39 categories of solid precipitation (including 35 different kinds of snowflakes), which can be further subdivided into 121 subtypes. And looking at some of those simpler shapes, it's clear that some snowflakes can probably look exactly alike — contrary to popular belief.

Why do some snowflakes shapes seem so different from the "classic" shapes many of us are used to? Certain snowflakes tend to occur only under certain temperatures and humidities — some of which might be more unusual in specific locales. Scientists have been able to confirm this general temperature/humidity trend by making artificial snow in the laboratory:

Further reading:

The Shapes of Snowflakes, at the Compound Chemistry blog, which has an interesting discussion of the whole topic

Some awesome photographs of snowflakes, including some of the odder shapes

Caltech physicist Kenneth Libbrecht has a really great site all about snowflakes, including FAQs and pictures