DisplayMate, the authority on display testing, used to have a Note 3 and iPhone 5s comparison but Apple forced them to remove it since it cast the iPhone negatively. In a nutshell, color accuracy/reproduction are equivalent while the Note 3 is clearly better in PPI, resolution, black level and size.
http://www.displaymate.com/Galaxy_Note3_ShootOut_1.htm
http://www.displaymate.com/Galaxy_Note3_ShootOut_1.htm
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