OP, I'm surprised that you think that the Note 3's pictures are amazing. I've seen the sample pictures compared to the 5s and they're crap.
I don't think I have said anything about the quality of the images, yet. I just shot side by side this morning at sunrise, a few shots, to test low light. I don't expect to be impressed with the Note3 images. I know the max ISO is 800 vs. 3200, but that in itself is not indicative.
Now, the things that ARE amazing include the over all interface of...the phone, the camera inter face, the ability to fire it with a voice command at a 2 second delay will help a good bit in getting better pictures for the average person. I know how to hold steady and have gotten some stunning shots with the iPhone.
Having taken well over 5000 images on the iPhone, won a major competition with an iPhone image, and built webpage tutorials around the benefits of iPhonetogography including pano, hdr, and timelapse, I am well aware of what I may give up. I also realize that there is a reason why I sometimes carry my Nikon D3, D800 or D7k. The fact of the matter is that I have taken some images that rival not necessarily the quality, but the spontaneity and unique composition possible with the iphone as opposed to haveing the better, larger camera around my neck. In fact, the image that won a contest last summer, against a gallery full of DSLR images, was taken with the iPhone held over my head shooting down into a flower I couldn't even see very well (a Magnolia) while two DSLRs hung around my neck. I would have needed a step ladder to use the other cameras.
The point is, I don't expect the Note 3 to be as good of a camera. Ultimately, I have decided that the main roll I want from my cell phone is to be a good PDA. I use the camera phone as documentation 9 out of 10 times. I shoot examples, I document, I geotag places I shoot with the full camera. Almost always, when I shoot a beautiful sunset, or fantastic vista, I have my full size camera or two with me. In fact, I have traditionally carried two iphones with one, an iPhone 4, dedicated to timelapse only. When I run around shooting stuff, the first thing I do is get out and setup the iP4 on a time lapse, and take an iP4s shot to immediate capture the scene. Then, I work the scene with a better camera, mixing in some HDR from the iPhone 4s, as well as some auto-stitch and panos.
I am going to do some more testing this weekend. Look over the tests I took this morning, but am prepared for the inevitable that the iPhone 4s will still live on as a camera aid to my photography.
In the mean time, the Note 3 continues to impress me with amazing features and capabilities.