iTunes is a music store where you can purchase songs and download them. Apple Music is a subscription-based streaming service. With iTunes, you pay to own a license of a song.
With Apple Music, you pay for access to a catalog of songs through a monthly subscription fee.
If this sounds confusing, let’s use movies as an example. When you buy a movie from iTunes, you pay a one-time fee and are granted a license to download the film. This resembles the traditional business of buying a movie. Now, if you pay for a Netflix subscription, you pay for access to watch movies but you are not paying to own any single film.
If one day Netflix decides to remove a film from their catalog, you do not have any way of reclaiming it because you do not own the license. This is pretty much the same for music.