I havent used a Lumia 1020 (the 41MP camera) in real life situations (tested at store), i watched many comparison online videos and although the 5s isn't directly comparable...it does a very good job of competing. It definitely snaps pictures quicker and the overall experience easier, whereas the 1020 seems to require strong focus and manipulation. I would personally say the 5s is the point/click superior camera device, whereas the 1020 is the focus/photographer leader.
I've personally owned the 1020 and I can comment on a few things here:
Nokia put out a really great camera for the 1020. The gymnastics it can do compared to the competition in order to capture an image is unmatched. Where the Lumia 1020 excels is in its ability to take an image and manage it after the fact. Now what do I mean by that:
Let's take the obvious example that Nokia has been using for their in-store displays and look at a shot featuring some hot air balloons. You take a wide shot, zoomed out showing 3-4 hot air balloons within your immediate area. Instead of snapping individuals photos for each balloon and crew, the one shoot captures enough detail for someone to zoom in on specific details of the photo and in effect create a new picture. Cropping a 1020 photo compared to some of the lesser cameraphones of 2013 and 2012 will show you how much better optics have become.
On the flip side, say you took a picture of your now ex-girlfriend and in the main shot zoomed in on her cheating, smiley face. Previous phones would have only captured the zoomed shot and nothing more. The Lumia 1020 captures the zoomed photo, however you can then zoom out and see the rest of the photo. The post processing that can then be done to remove your ex-girlfriend and focus on the rest of the shoot in-app is wonderful.
It is a shame that the 1020 is hampered by an OS that at times is too simple for its own good and a convoluted series of chained camera applications (Lenses) that opened slowly and had the potential to result in missed opportunities and photos that were out of focus or were taken with a lens that doesn't have the full 41MP oversampled image to manipulate. (Because of speed concerns)
If you are taking photos of subjects with the 1020 in a controlled environment and have intent behind your shots, the results are very good. If you are casually snapping up pictures of everyday events you may come away disappointed or have fewer "keepers" because of the shutter lag and post-processing done on the Lumia to process the image you just took.