The difference in the tileset you're referencing, and the red colours you can see in NEXXT, is probably because the two have been using different emulator palettes.
Some background:
On the NES, there is no 'true' RGB interpretation of its system palette, because it's generated in the analogue domain in a wholly different way. So instead, you have all these different emulator palettes that are different persons' interpretations of what they wanted the colours to look like.
Some are worse than others in terms of accuracy, and it can be difficult to trace the internet genealogy of bad palettes. FCEUX standard palette is one such inaccurate palette, though, which has some strongly misinterpreted colours, including a really vibrant slightly brownish red that mismatches the vibrancy of the rest, and that just doesn't exist on a real NES, among other problems. I can only guess, but that could be a possible source of the red you've seen in the reference tileset.
Now, some hands-on tips:
If you want to replace the system palette in NEXXT, take your favourite palette from an emulator (a .pal file, should be 192 bytes large), rename it "nes.pal", and place it in the same folder as NEXXT.exe. The next time you start it, it'll use that palette instead of its internal one.
Some personal recommendations are unsatured v6.pal and fbx_smooth.pal, which are balanced, and often bundled with popular emulators.
Sometime in the future, i'm going to make this process smoother by having a built in system palette browser, but i can't give a timetable.