I get 31mb down and 11mb up when tested in both scenarios. That's the actual throughput speed of my connection here at home at all times and I have always had that speed on the Note 3 (shows same on my firewall) .
I used to get the "unstable internet connection" message, but I changed my DNS server configuration on my WAP's and that problem went away. It wasn't a Note 3 WiFi issue and I am pretty it's generally not, in any case.
I don't think the speed has anything to do with WiFi power save mode. WiFi Power save may affect other things, but in my case, it has nothing to do with download or upload speeds.