Just a run down on the options on both
Screen size 5"(M8)/5.7" (Note3)
Screen resolution Both 1920x1080
So PPI is a little bit lower on the Note 3. I would question if you can really see it
CPU Snapdragon 801(M8)/Snapdragon 800 (Note 3)
GPU Adreno 320(M8)/Adreno 330(Note3)
Ram 2GB(M8)/3GB(Note3)
Camera 4mp(M8)/13mp(Note3)
Speak sound Stereo Boom Sound(M8)/Crappy Single Speaker(Note3)
HD Recording 1920x1080 Maxk(M8)/UHD 4k(Note 3)
Starting Storage 16GB(M8)/32GB(Note 3)
From a pure spec perspective, the only thing the M8 really has over the Note 3 is the Upgraded CPU to the 801 model, and Boom Sound. The cpu isn't really that much of a improvement because the Note3's Snapdragon 800 runs faster then most other phones with the same CPU.
The only reason to select the M8 would be for boom sound or the fact you like the overall design. It isn't a bad phone but the Note 3 is just really a great phone across the board. I really don't think there is anything better out right now overall. At this point the only reason I would give up the Note 3 is if I really just couldn't take the size any more.