Obsidian is silicon-rich rock that cools so quickly that it does not form crystals. But glass -- obsidian -- is not chemically stable. Snowflake obsidian is obsidian that has started to degrade, and so has formed spots of crystals.
Obsidian has a tendency to slowly morph into cristobalite, a process accelerated in the presence of water vapor. This forms small clusters of white crystals to appear in the obsidian, generally spreading out from a central point. In its unpolished form this looks quite a bit like white splotches of lichen, but when polished... it still doesn't look much like snowflakes. Still, the contrast between gleaming glassy black and snow-bright white splotches is striking.
Cristobalite is a transitional stage of obsidian crystallization, and in few million years the obsidian will morph into quartz, tridymite, or perlite. Leftover globs of obsidian found in perlite beds are sometimes called Apache tears.