I think Apple Music's "New Music Mix" is their version of Discover Weekly, it's under the Listen Now tab

Yeah - it sucks in comparison. Not at all similar to anything in my library. It might as well be random.

The New Music Mix only releases one playlist a week. I believe Discover Weekly releases five playlists a week, each with their own moods. I much prefer Spotify's approach.

However I do find that both services largely recommend the same music to me. They just have different mechanisms of surfacing it.

Discover Weekly is once a week: on Mondays. You're thinking of Daily Mixes, of which there are 6 that update throughout the week, one playlist per day.

To go along with Discover Weekly there's also a Release Radar playlist that updates every Friday with that week's new releases from your artists.

Spotify is so deeply engrained into my weekly schedule that I really look forward to the Discover Weekly and Release Radar playlists bookending my work weeks.

That's fair, same for me. Discover weekly (and to an extent release radar) were some of the best features on Spotify for sure, they had a way higher chance of really hitting good than they have any right to.