OK so I think I may have figured it out.
I noticed that if I kept reading on my iPhone and then moved on to the Paperwhite I could force the Paperwhite to sync by going to the menu and hitting "Sync to Furthest Page Read". It would then show me how far I had gone on my iPhone and would ask if I wanted to sync.
But if I then kept reading on the Paperwhite and got back again to the iPhone, then the iPhone would not sync to the furthest page no matter what I did. It was as if the iPhone didn't know the Paperwhite existed but the Paperwhite would acknowledge the progress I made on the iPhone.
I've tried deleting the Kindle app and reinstalling but nothing changed. I rebooted the iPhone and the Paperwhite, still no go.
So I tried something different, I went to Amazon's
Manage Your Kindle page, looked for the iPhone under "Your Devices" and hit "deregister". Then deleted the app on the iPhone, reinstalled it, reregistered my device, downloaded a book and... voila! It asked me if I wanted to sync to the last viewed page on my Kindle.
So far so good but I'll test with a couple more books before claiming victory. Thank you all for your input.
