I have a similar issue:
As far as I can tell it started after installing iOS 9.0.2 on my kid's iPods a few days ago and the iTunes app is now broken. Installing 9.0.2 was the last thing I did with iTunes, now it won't download updates, it says it's connecting to the iTunes Store, then does nothing....no errors at all. It also won't sync updates downloaded using the iPod. It goes through the syncing motions, but doesn't actually do anything and doesn't return any errors. This was on iTunes whatever version of iTunes I had previous to 12.3, on Windows 10 which was working before and for the iOS update, so I figured perhaps I needed to perhaps install the new iTunes app, no go, same issues.
After some experimentation, I discovered starting iTunes as administrator seems to get downloads going for me, but syncing stuff from the iPod to iTunes appears to still be broken. I have not tried the other direction as I've already downloaded all available updates directly to the iPods. So, I can redownload the updates using iTunes, but I cannot sync the updates from the iPods I've already downloaded. Either way, I've never had to run it as administrator. Typically, I just plug in the iPod, iTunes pops up and I do updates. I wonder if there is something new in 9.0.2 that has to be set to facilitate this.
That said, even with Administrative mode enabled, it appears to refuse to download individual updates. I tried two different apps, Dragonvale and Candy Crush, I think. It does what it did as described above, with no error. It seems querying the update server for individual apps is broken, but pressing the update all button puts all the updates in the update queue and starts downloading.
Run Diagnostics for Network Connectivity says there is an issue with connecting to the itunes store on a secure link and that iTunes is not enabled in the Firewall, but it is enabled in the firewall and in fact has been set up so that it can accept inbound connections of any protocol on any port. That said, the secure link error seems to be a red herring. Turning off Windows Firewall altogether does not get rid of the error: so the network is theoretically exposed for all traffic incoming and outgoing on all open ports/protocols. It would appear to be an SSL connection issue on Apple's end.
Everything seemed to be working fine a week, or so, ago with the previous 8.x.x version of iOS and the iTunes that was there previously to upgrading to 12.3.