ITunes is a media player, media library, online radio broadcaster, and mobile device management application developed by Apple. If you want to download music, movies, audiobooks, etc., in your iTunes library, you need to purchase them in iTunes Store. Once purchased, you'll be able to keep them forever.
Apple Music is a separate music service provided by Apple company. It is comprised of music and video streaming service. The entire music catalog of Apple Music exists in iTunes. Users can select and stream music to their device on-demand, or download them for offline listening.
More importantly, music purchased from the iTunes Store after 2009 are recognized as files without DRM protection, which means you can sync them to other media player or devices. However, all the contents on Apple Music are still DRM-protected. Once the subscription expires, users no longer have access to those even downloaded files. To keep Apple Music forever as iTunes music, you can borrow help from Audfree Apple Music Converter.