We have determined that Apple Music songs may appear in Anytune (if saved for offline playback), but cannot be played by Anytune.
If you are using Apple Music, we recommend that you disable the iCloud Music Library and sync any songs that you want to use with Anytune via iTunes sync instead.
This is because songs downloaded to your iPhone or iPad from the iCloud Music Library or Apple Music are DRM encumbered and can only be played back by the built-in Music app. iTunes Match uploaded songs are still OK as iTunes Match downloads are DRM free.
Needless to say, this is quite disappointing for users of Anytune or other 3rd party music players.