For those who can see the playlist in the Music app but cannot see the contents of the playlist.
Force quit the Music app and relaunch it.
1. Press the home screen to ensure you are out of the Music app.
2. Double tap the home button.
3. Press and hold the Music app icon in the running apps list.
4. Touch the red - to force close the Music app.
5. Press the home button to exit.
6. Launch the music app again.
I had the podcast playlist problem. My Music app now functions mostly as it did before iOS6. I have also setup a new iPhone 5 and it is working as well.
I do only use smart playlists so cannot confirm if this works for manual playlists.
The full fix that I did is as follows.
1. Delete the podcast app from my phone.
2. Shutdown and Restart the phone.
3. Force quit the Music app as above.
4. Podcast Playlists now show up in the Music app and all of my podcasts are listed in the order that I had set them up in iTunes. Syncing and updating of the playlists is also working as before.
I hope this helps some of you.