I was quite surprised when I upgraded from the 4S to 5S in the camera improvements.
It's hard to explain but the photos seem to come out sharper and with better more accurate color, IMO.
This is my experience exactly.
The reality of the camera is that it is an incremental upgrade. Keep in mind that these are still cell phones first so expectations of camera should be inline with that. The 5S camera adds slow motion video (120fps @720p) which is kinda neat, burst mode (10 fps) and dual LED flash as new functionality over the 4S. Other improvements are a faster aperture (f/2.2) and bigger photo sites on the sensor which should improve low-light sensitivity.
The quality of the stills will not be noticeable in the majority of your outdoor or well-lit pics. The differences will come in low light with or without flash. The new dual LED flash does indeed improve color accuracy in low light with flash but if you're like me you avoid flash at all costs so you'll rarely see this. In other low light pics it is indeed better than the 4S (when not using flash) but don't expect massive improvements over the 4S.
I would like to respectfully disagree with you here. I think what you are describing would fit better with a description of the iPhone 5's camera vs the 4S's. In my opinion, the 5s is in a realm entirely above both of them.
I have noticed that colors vibrantly pop with the 5s and that everything looks much more natural. Image quality that previously could only be achieved with a 3rd party image editor or HDR app are now achieved natively on the 5s's camera.
Take, for instance, these two photographs. They are both on the stock iPhone 4S and iPhone 5s camera, and I took them. HDR is on. The one with the sun visible is from the 4S, and the one with the sun behind more clouds is from the 5s. You can see that they both do a great job - however, the sky color is much more vivid in the 5s's picture, and there isn't yellow saturation like there is with the 4S. Also, the houses along the side that are hidden in shadow with the 4S are brighter and more visible with the 5s.
Sorry for the overall reduced image quality - I took these photos off of my Facebook.
The 4S's camera is by no means a slouch, but it does get beaten out by the 5s in my opinion. The 5s lets in way more light, which is great in all situations. And I agree with you that while I don't like using the flash, I am less hesitant to do so with the 5s's True Tone flash.