Hi Sheila, I bought an iPhone 5s and handed my 4s over to my daughter.
I setup the 5s from the backup from the 4s and saw the same problem.
I unsynced the music and had the same issue as the smart playlist contained podcasts and was therefore not removed.
I tapped on the podcasts tab in iTunes and iTunes has frozen (this is an issue mentioned in other threads and the only cure is to disconnect the internet - https://discussions.apple.com/thread/5339836?answerId=23099713022#23099713022 ). So, I disconnected the internet and iTunes responded again so I was able to switch off podcast syncing.
I then turned podcast syncing back on and resynced to the 5s.
On the 5s I went into the Podcasts App and My Stations:iTunes Playlists:Podcasts(a smart playlist containing my podcasts) and lo and behold the two most recent podcasts were still missing.
This has now taken me a fair bit of time and also included the purchase of a new iPhone (which I was going to do anyway).
This is a bug in iTunes and I would suggest you submit it as such.
The frozen podcast tab is another bug which also needs notification.
I am an iOS developer and my customers depend on a reliable platform (which includes iTunes) as my app is used for backing tracks for stage performers.