I have a different music library on my PC as on my ip5

I change the music that I listen to on my ip5 about once a month. Sometimes more often than that. Recently I´ve gone to my itunes library on my PC and once checking or unchecking the sellected songs that I want to then sync onto my ip5, I´ve clicked on "sync" on the PC and wait while the sync process is underway. The PC is online and the ip5 is also online and connected with USB. Once completed, I think everything is as what I wanted, but when I look at what is on my ip5 in the music library, there is music that should have been removed and still listed. I´ve re-synced again a few more times, but I´ve still got some of the music that should´nt be there. I´ve also noticed on my PC that some music that I´ve purchased on the ip5 and later transferred to the PC have a small broken circle infront of the title as if it was being downloaded. Could the problem exist from the fact that when I plug my ip5 onto my PC it seems like ages until it prompts and then a small window pops up asking me if I trust this computer, which is of course my PC.