Both Pandora and Spotify are free music streaming services that allow users to listen to music online without having to download or store songs. Pandora uses a music genome project to create personalized radio stations based on songs and artists input by the user, while Spotify allows users to search specific songs and playlists. Pandora offers ad-supported free listening with premium subscriptions available, while Spotify has both free and premium subscription options. The document provides details on the origins and features of each service.