A snow crystal, as the name implies, is a single crystal of ice. A snowflake is a more general term; it can mean an individual snow crystal, or a few snow crystals stuck together, or large agglomerations of snow crystals that form small balls that float down from the clouds.
Snowflakes are not frozen raindrops. Sometimes raindrops do freeze as they fall, but this is called sleet. Sleet particles don't have any of the elaborate and symmetrical patterning found in snow crystals. Snow crystals form when water vapor condenses directly into ice, which happens in the clouds. The patterns emerge as the crystals grow.