I think Apple Music's "New Music Mix" is their version of Discover Weekly, it's under the Listen Now tab
However I do find that both services largely recommend the same music to me. They just have different mechanisms of surfacing it.
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.