Apple Music is the name for Apple's music subscription service. It can also be used to refer to the Music app on computers running the macOS 10.15 Catalina operating system and iOS devices. In Catalina it, and new apps for TV, Podcasts along with Finder and Books, take over the functions of iTunes. See What happened to iTunes? - Apple Support for more.
When you sign into iTunes (or Music) with your Apple ID on a computer, iOS device, etc. you will have access to locally stored content and any unhidden purchases with cloud links from which they can be streamed or downloaded. iTunes Match extends this ability to stream or download to the rest of the music in your library. Apple Music goes further and lets you add other content from the iTunes Store to use in the same way for as long your subscription remains active.
iCloud Music Library is the older term for Sync Library. Turn iCloud Music Library on in iTunes from iTunes > Preferences > General tab and on your device. Any content you add to one device should appear on the other. In general life is easier if you have a single Apple ID, but having different IDs for iTunes (and related services) and iCloud shouldn't cause a problem as long as iCloud Music Library (iTunes Match or Apple Music) is attached to your iTunes Apple ID.
tt2