This is because when Playlisty creates your playlist(s) it creates them in the cloud and (usually) it takes a little while to sync back to your device. There’s nothing you …
Say there’s a playlist that gets regularly updated on an external service such as SoundCloud or Spotify e.g. perhaps a friend shares their favourite tracks in an “On Deck” playlist …
“Save to playlist” does what it says: it saves the tracks you’ve selected to a playlist. “Save to library” will not save the selected tracks to a playlist. Instead it …
“Save to new playlists” will save whatever playlists you’ve selected to new playlists, including your “Liked Songs” and “Liked Albums”. Or in other words: Playlists -> Playlists (with same names) …