Think of Pandora as a radio station, except the DJ knows exactly the type of music you want to listen to. You give him an artist, or a genre, and he just goes off and starts playing songs. Some you'll have heard before, others you won't.
Spotify is more like playing your songs on your hard-drive. You pick a song, or an album, or create a playlist, and off you go. It'll play that. There's no randomness, you won't discover stuff just by playing music and it won't recommend new stuff for you.
However, using Spotify you can link in with your Facebook account. This is awesome - you can send a song, or an artist, or an album to your friends. Or you can put a link on Twitter or Facebook. Anyone else who uses Spotify can then open this link and immediately listen to the song.