Unless I misunderstand the article, all Apple has the exclusive rights to is this human curated playlist. The tracks within the playlist are still available on Spotify, and I'm sure somebody will write a bot that looks at the playlist on Apple Music and produces it on Spotify each week, so you could just follow that new copy-bot playlist instead, if you cared.
I'm guessing that for this to really work there must be some exclusivity in terms of songs on that list. Ministry will own a lot of the licenses for these songs so maybe they will stagger releases between Spotify and Apple. That way, if you really want the complete list of new music as curated by Ministry you will have to be on Apple.
Also, doesn't this tie in with law suit that Ministry were pushing where they were upset that people were copying their compilations track lists as Spotify playlists? I don't even think they won that one.