I imagine this is one of the benefits of installing software on your computer rather then streaming on your browser like you would on amazon or google. I doubt bandwith has anything to do with it. I haven’t noticed much difference when using the mobile apps for all three services.
Spotify uses a combination of peer-to-peer and client-server for downloads along with a generous-sized local cache. If you’ve played the song recently, it will pull it from the cache.
If it is not in your cache, it can grab the first part of the song from either the Spotify servers or from another user with that song in their cache.
A big part of their early focus was getting that user-experience just right so that playing songs on Spotify was like playing songs locally.