As noted earlier, I have an iPhone 6+ with 128GB and am using the most current OS and iTunes. I was unable to sync any media -- music, video, tv shows, audiobooks -- to my new phone. But I was correctly syncing apps (and books, I think). Everything had synced fine previously on my 5S. I tried everything suggested here (except there was no media on my phone to delete per johnee1010) short of the complete factory restore. I tried rebooting the phone and my Mac. Nothing worked. So I resorted to Apple Support Chat, who took a look at my iPhone Summary page in iTunes, and had me attempt to sync a music track while they watched my screen. It failed, of course. One thing to note -- which may or may not apply to others here -- iTunes listed my phone number on the Summary page as "n/a", rather than reporting my correct phone number. Something others may want to check.
Chat Support told me to perform the dreaded complete factory Restore (the one at the very top of the Summary page). So, reluctantly, I did. It took hours to complete, including downloading a new copy of the OS. I elected to restore contents from my most recent iTunes (not iCloud) backup (not sure if this was actually a good idea). System restoration complete, the phone lit up with "Hello", etc, while at the same time iTunes said it was restoring from backup. iTunes copied over 6 apps and then (apparently) choked on the Find My iPhone app. It simply sat there and reported it was copying the seventh app (a small one), for about a half hour.
At this point, given the problem with the Find My iPhone app, I had a nagging suspicion that restoring from backup *before* the phone's initial setup was complete might not be a good idea (that's what I did the first time, on Friday). So I hit the X on the Restore progress bar and I went through the initial setup on the phone, including signing on to iCloud, etc. (This is a second point others may want to pay attention to.) I hit Restore again and copying from backup resumed immediately, no questions asked. When it was done, everything was loaded properly and media now copied over correctly. So far, media continues to copy correctly.
It took quite a few hours to go through the entire routine, but it worked. As onerous as a factory Restore is, others who continue to have media-copying problems with new iPhones may want to try this, paying particular attention the two noted points above. Your mileage may vary, of course. (Note that I have a neighbor who experienced the same problem with a 128GB iPhone 6, not 6+. He was able to fix the problem without resorting to a Restore. In contrast to my situation, however, iTunes did correctly report his phone's phone number.)
