You found the iPhone 5s camera that much better? I have been using both. The Note 3 seems much better in good lighting and with video (the video clarity on the Note 3 is phenomenal). The iphone5s takes the better low light pictures.
Colors are not as accurate on the N3 and the white balance is usually off. You have to hold it VERY steady or you get blur in pics not in bright sunlight. Video is a bit choppy when panning around (a huge issue every Android phone suffers) and the sound in the videos is not nearly as rich and clear as it is on the 5s. The one place the N3 does excel is all the modes and ability to manual control many settings.