I had the same problem and was able to resolve it.
Please note that I was pretty much at capacity before trying to upgrade to the latest software. I assume the new software update was larger than the previous version which caused the error trying to restore all the data from before but not having enough room now.
On the iphone, go to Settings > General > Reset
select "Erase all Content and Settings"
Connect to iTunes > you'll see "Welcome to Your New iPhone" page.
Select the "Restore from this backup:" and the restore commences.
After the restore, the phone will reboot and return to the "Welcome to Your New iPhone" page. Be patient. It will take a moment. Select the "Setup as New Phone" - your iPhone will sync all apps and music.
All Apps and data were restored successfully! (Took awhile though)