Ok. So as an amalgum of several responses, I did get this to work, finally. First off, my account was a local administrator account, so the admin pieces as to why or why not the driver installed did not make a difference.
This seems to work for Windows 10. We have had issues with the "Apple Mobile Device Support" in the past. Not the same issue, but this made me look at that.
What I did to make it work was just to uninstall iTunes and the Apple Mobile Device Support program.
Downloaded a fresh installer from Apple for iTunes and ran that.
iPhone was recognized on first plug in after that. Didn't even have to reboot.