I had the iPhone 5S and I didn't find the camera to be amazing. The pictures were ridiculously post-processed to the point that a lot of detail was unsalvageable in image editing. My friend's S4 took sharper images, and I'm sure some of this had to do with the higher MP count - however, I did examing the images in 1:1 size on a computer monitor not just on a phone screen. I find the Note 3 to be as good as the S4.
The only area where the iPhone pulls ahead is in low light, but the Kit Kat update for the S4/Note 3 is supposed to come with a Camera FW update that improves low light photography noticeably so the jury is still out on that for me. I don't take much low light pictures, anyways. I still find the Flash on Samsung devises to be better than the iPhone 5S. I didn't really notice any improvement from the "True Tone" flash. In most cases, Auto-Correcting (especially using decent desktop computer software) the image from another phone gave a better skin tone, but without the camera's aggressive post-processing destroying image quality in other areas...
The pictures looked amazing on a smartphone screen, but when I loaded them up on my PC to crop some pictures (I do a lot of cropping) I noticed people looked like they were painted (Very soft edges and a lot of facial detail was smudged/processed out and a ton of background detail was smudged/processed out as well. There's nothing you could do in Photoshop or Lightroom to recover detail the camera software destroyed in those pictures...
Also, the 32GB iPhone I had with iLife, iWork, and a few other 1st party apple apps installed (Find My Friends, iTunes Remote, Apple Store, iBooks, Podcasts and maybe one or two others) had less storage than my 32GB Note 3 with all Verizon Bloatware Disabled, a bunch of installed Google Apps (including some from Play Store that weren't pre-loaded), and Samsung Apps installed (ditto, e.g. their Video Editor is a 120MB+ install and I added that

, I disabled Samsung Link however)... The Note has 23.08 GB free space in Internal Storage. The iPhone had < 21GB free.
So, after looking at things from the other perspective, I suddenly didn't feel as bad about what they put on these Samsung devices, Lol. Apple's iWork and iLife apps by themselves are like a 1.5-2GB install size on the iPhone. It is *much* harder to deal with a 16GB iPhone than a 16GB Galaxy S or Note Device, because there is no SD Card slot to Offload things like Camera Data (Pictures, Videos), Music (Play Music can Cache to SD Card now, for example), Documents, etc.
Totally Satisfied with the Note 3 camera. Have no issues with the phone storage or what Verizon puts on it, as long as I can disable it and never be bothered by it again
Another camera thing I noticed... All these phones coming out with OIS, but apart from the Lumia 1020, even those "low light specialist" cameras have been underwhelming in quality. Both the Lumia 920 and HTC One have been rather underwhelming. While they do let in a lot of light, even their low light images are at a lack for detail, to the point that I'd prefer "Low Light Mode" images from my old Galaxy S3 over the pictures I've seen/been sent from those devices. The LG G2 was said to not be as good as the Note 3 in low light photography or videography, despite having OIS, by some reviewers. OIS is nice, in theory, but I think that needs a little maturity before it becomes as must have as some people are making it sound like it is.