Purchased music is music to which you have purchased a permanent license. As long as Apple had a copy in the store of the track/show/movie you bought, you can redownload it if you do not wish to keep it on your computer (if you change country you will not longer have access to the same store and you may not be able to redownload). Note, however that sometimes a version will be pulled from the store. It may not be returned, or it may be replaced with a different version. In that case it will no longer be available for redownload. As long as you have kept a copy on your computer or an archive drive, it will play. There is no DRM protection on anything released in over a decade.
Apple's Music subscription service lets you listed to anything in Apple's collection for as long as you have a subscription. You can have copies of the material locally on your device storage for offline listening, or you can listen to the streaming version (which requires an Internet connection). If you let the subscription expire, the tracks will no longer play.
So with either you have the option to listen to the music with a locally stored track, or stream it from Apple (which requires an Internet connection). As far as Bluetooth goes, that will be from your device locally to your headphones. You do not have a Bluetooth connection to Apple.
Purchased music has the advantage that as long as your are good about keeping a copy and backups, you will always have access to the music even if Apple were to disappear. However, you are paying per track.
Apple Music subscription has the advantage that you have access to millions of songs so if your tastes change you are not stuck with dusty old music. Apple Music also has a Match service built in. Match lets you upload copies of your songs from all sources that even Apple may not have and have it available for streaming or downloading from iCloud.
