I have the Nexus 5, and have been overall very happy with it. Very good LTE reception, decent battery life, great looking screen, fast processor... Yeah, it's hard to beat for the price.
I used to have the Note III (and before that, Note II). I don't miss it nearly as much as I thought I would.
There are phones out there that do some things better, to be sure, but you'll be paying $700 instead of $350