Hey Stevo2905,
I became a member of Apple Support Communities just so that I could answer your question 😁, because I had the same one and couldn't find the answer (or anyone else besides you asking the question) anywhere. Hopefully my answer makes sense.
The problem started for me after updating my iPhone to iOS 10. After chatting to Apple Support for a few hours in total, the solution for me was to backup my phone onto my computer through iTunes and then restore.
After the backup and my phone being restored to setup, I tried to reorder the songs in my created playlist on Apple Music again and it worked. So I then restored my phone from the backup, getting all of my info and data back, and the issue no longer persisted.
The support Advisor who helped me said that it was probably a corrupt back up (after updating to iOS 10) and that the software may not have been responding properly because of it. And they did notice that I was missing a few updates while the restore process was happening.
After trying a few things, this is the only solution that worked. I know this seems like a lot to do in order to fix such a small issue, but if you use Apple Music as much as I do, I'm sure it's annoying to not be able to reorder your songs the way you would like. It shouldn't take that long to do, especially if you already know how to backup and restore through iTunes, unlike I did. But if not, the technical support members at Apple are great. I used the "Chat" option and they were able to see my screen and walk me through everything.
Good luck! I hope your playlist rocks!