I had the same issue. Very frustrating.
I found this explanation the best:
By selecting the iPhone button on the upper right in iTunes, Click on the Summary tab if it is not displayed already, scroll down to Options and then select Manually manage music and videos. This allows you flexibility to move individual songs and lists onto the iPhone.
Also, under the Music tab, you can change how iTunes synchs with the iPhone. The video explains both these options...
This did not help. When I drag playlists from my "playlist" menu directly to my phone nothing happens. When I go to "music" tab on the phone the only thing on the screen is "sync voice memos". There is no selection under the summary for me to "manage music". There is one for "manage videos". I have tried turning on the "view side bar" which also did not allow me to drag songs to my phone.