240p allows every scale factor (1x,2x,3x,4x,5x,6x,7x,8x,etc) while 720p is limited to 3x factors (3x,6x,9x).
If you want to connect to an external scaler, this is bad. If you want to connect to a PC capture card and play on a PC, this is bad. If you want to connect to a CRT this is bad.
720p scale factors do not fit nicely into 1280x1024,1680x1050,1920x1080 or 1920x1200, 240p scale factors do. Even if you don't want integer scale factors, more options is more flexibility to trade between sharpness and image smoothing in the scaling.
It's a very weird decision not to be able to output native resolution. Of course this is only relevant to people obsessive about image quality.
4.5x nearest-neighbour scale (emulator, hi-def NES, maybe framemeister?):
You shouldn't be using uneven scale factors as a basis for comparison, best image quality from a Hi-Def NES or XRGB-mini comes from 4x (small black bars) or 5x (some overscan, XRGB-mini only). Why? Uneven scaling introduces artifacts, from your image:

All those pixels should be equal size. They're not.