The Note 3's camera is much better in daylight and it records 4k video.
And if you're using Nova launcher and Widget locker.. Why not get a Google Edition S4?
I hate TouchWiz but I love the Google Edition S4. I own one.
The Galaxy S4 is an outdated phone. It's like getting an iPhone 5 now.
I also had an S4 and even then, it wasn't up to snuff. Screen, speed etc.
Finally, the Note 3 seems like the first Android device that doesn't have a fatal flaw (still AMOLED but much brighter and more useable). Plastic but rear looks and feels nice (and has the benefits of removeable battery/storage). Acceptable camera, fast SOC and enough RAM (barely, actually. With 3GB, Android only allows me 2.26 or so, and most of the time, 1.5gb is being used with the basic apps running like Flipboard and Gmail. Android appears awful with memory management).
Only real problem with the Galaxy Note 3 is that it's a Samsung Galaxy Note 3 and not a Google Nexus Note 3.