"Apple Music" integration in Amazing Slow Downer lets you playback songs from all of your existing "Apple Music" playlists and the songs you have in your "Library".
If you miss a song, just switch over to the iOS Music player app and add the song to the library or one of your playlists.

The songs in your music library may be of different types. They may be Apple Music songs stored in the cloud and streamed during playback. Or they might be stored as physical files on your device. In this case they might be DRM protected or regular unprotected files. DRM is an acronym for Digital Rights Management which here means that the files can not be played back in the same way as regular audio files.

There are some limitations for DRM protected songs and songs stored in the cloud being streamed during playback:
The audio quality is not changeable and is somewhat lower than in other sections of Amazing Slow Downer.
The "Pitch", "Mix" and "Equalizer" functions are not available. "Change Pitch at New Loop" and exporting of audio is also not available.
The reason is that it's not technically possible. All other player functions in the app are accessible.

Regular unprotected files stored on your device doesn't have any of those limitations.