"SpotPass" refers to the Nintendo 3DS's ability to access a Wi-Fi connection in order to automatically download certain kinds of content. "StreetPass" refers to the Nintendo 3DS's ability to connect to another 3DS system and swap certain information (also wirelessly, though without the need for a Wi-Fi connection).
SpotPass is generally used to download demos, videos from the Nintendo Video Service, SwapNotes, and additional content for games that you already own.
StreetPass allows two Nintendo 3DS units to exchange certain information. This information includes Miis (collected Miis will automatically go into the Mii Plaza), game-specific features (like Super Street Fighter IV: 3D Edition's trophy-swap function), and SwapNotes.