Given its popularity, it sure as well took long enough, but Spotify has finally been updated with support for iOS 7. The update brings a subtle yet certainly noticeable redesign that incorporates flatter textures, and divided search and browse sections.
Spotify's release notes were a bit ambiguous, stating that the app was "now compatible with iOS 7." Since compatibility doesn't necessarily guarantee any sort of redesign, I wasn't really expecting much.
Thankfully, Spotify's definition of "compatible" goes beyond the codebase and encompasses the look and feel of the app as well. You'll quickly notice new visual cues, which provide a more cohesive look alongside iOS 7.