And the winner is neither.
I own both these phones and use them extensively to take photos.
Below is a brief overview of each and what you should expect.
Speed – From startup to capturing the moment goes to the iPhone 5S. And then the 5S burst mode of 10 photos per second at full resolution all the way up to 999 photos as a burst is out of this world. I’ve tried it and it works. It does take a while longer than normal to analyse 999 photos and give you the best but it’s still quick. Meanwhile the Note 3 only takes 20 shots on burst but it’s very quick once it starts. From holding the shutter button down seems to have a delay of 1- 2 seconds before the burst mode starts.
Panoramic – this goes to the 5S as it auto adjusts exposure as you pan. But both are good at this mode. And both can take some fantastic shots. With the Note 3 be sure to hold phone vertically to get a real large photo.
Low light or indoors – this is a close draw. Both have strengths and weaknesses here. The Note 3 uses the flash more often whereas the 5S tries to avoid using the flash. On scenic non moving night shots the Note 3 using a tripod will outperform the 5S.
Good light – both take excellent photos. The 5S has an uncanny way of making sure the photos are very natural in their colour reproduction. But the Note 3 has higher mega pixel lens. And it shows. I actually prefer the Note 3 results slightly.
Macro – a close match. But the Note 3 wins. It’s one of its strengths.
Software – the 5S is limited and is really a “just press the shutter” whereas the Note 3 has comprehensive options. Depending on your needs depends on your winner.
Overall I prefer the Note 3 photos but I do miss that speed to first shot that the 5S achieves.