See if this is what you are doing.
Start iTunes on your computer.
Connect your iPad to your computer.
Select your iPad in iTunes on your computer. If you don't see your iPad listed use the View menu in iTunes to Show Sidebar.
Once you select the name of your iPad a row of tabs should appear to the right. Select Music.
Under the Music tab select copy all music or select only the playlists you want on the iPad.
Then click on Apply or Sync.
If this is exactly what you are doing please explain where your iPad or iTunes varies from what I describe.
I have an iPad 3 and it is doing the same thing. Seems to be an issue with iPads. 5S has the play lists correct and so does my old 3G that I use as an iPod now.
My playlists won't sync up to the way they are in iTunes. I have to manually put the songs that aren't in my playlist on my iPad in.
Music won't download to my iPad. Shows the little cloud until I click on the cloud to get the music to download. My settings are to do automatic downloads of music.
I tried doing the power restart and then syncing up again but no change. I tried plugging in to do a sync and tried via WI-FI. Nothing gets the playlists to sync correctly.
I also have my iPad set up to copy the enitre library from iTunes.
Then I see this post was from Feb. and no response so guess it never got fixed.
I am having this exact same problem. Ipad 3 appears in iTunes, I select manual selection of music and videos, sync, and the new selection of music playlists don't appear on my ipad. Moreover, the videos pop up for a second, then close. One video, which is about 81 minutes long, reads as 1:20 on the screen - ?!