I have both. 5S is my primary device. Note 3 has many things to like, and it serves me well as a WiFi phablet around the house. I wasn't as happy with the camera in low light or moving objects situations. As others have stated, the 5S camera excels all around.
While I generally like TouchWiz, there are a few bloat ware problems with the Note 3. Hard coding My Magazine to the Home Button was a real disappointment. Short of using a third party launcher as a workaround (which defeatured a few native TW options), or rooting; there's no way around that. Apple doesn't force bloat. Other apps get pushed and updated without any control over the process by the user.
Another area to consider is integration with your computing and communication ecosystems. It's easy to collaborate and communicate with others in the iOS world given all the features that iOS 7 provides. If you're a Mac user, then the integration is seamless.
iOS is not perfect and lacks a few features I'd like to see; but overall for me the hardware quality, timely updates, improving iOS features and functionality, and fantastic support are good reasons to stay with the 5S and iOS.