This may be semantics, and I am not sure what counts as Apple Music versus iTunes these days, but you can download all files for offline listening in Apple Music. I assume if you can download a lossless file from Apple Music that you can also buy that same file in the iTunes store and then download it.

They both use the same app for playback, so this is a tad confusing.

Yes, but AFAIK things you download via your Apple Music subscription are still DRM-protected and can only be listened to as long as your subscription remains active.

Whereas the iTunes store downloads are a one time-purchase and have no DRM, so they're yours to keep.