If you're syncing your iPhone to iCloud then you cannot edit playlists directly on your device, you can only edit playlists in your library (which in theory gets pushed to iCloud and then to your device).
But are you syncing to iCloud, or manually managing your iPhone, or auto-syncing your iPhone? Your symptoms sound like you're auto-syncing, in which case you cannot edit things directly on your iPhone either and have to do all editing in your library, then re-sync to apply the changes to your iPhone.
What I described above works for playlists in your library, and if you have the "Manually manage music and video" option selected for your iPhone you can do the same directly on your iPhone while connected to iTunes.
Since you're not able to make edits on your iPhone, you must be either auto-syncing or syncing to iCloud.