I have a Rebel XT (same sensor as the 20D, or very similar) and a 5D III.
The difference in image quality is not as great as I'd expected. At a given ISO between 100 and 1600, per pixel, the cameras perform within a stop or so of each other, which is surprising given how new the 5D III is and that they have similar pixel density. As regards sharpness, the Rebel is about as good for a 8x12 as the Mark III is for a 12X18, I would guess, and at the same print size the 5D III probably has over a stop advantage in terms of how clean the image is, nearly two stops. It has a bit more dynamic range.
For some bizarre reason the Rebel XT seems to handle JPEGs better, while the 5D seems soft per-pixel. I don't know why.
The Mark III has a beautiful finder, amazing fast autofocus, great build quality, and a nice LCD. Full frame gives a really special look with fast 35mm and 50mm lenses (and 135mm lenses, too, I think). I think you will not use your 20D much after getting the Mark III. It's a very nice camera.