@Neuroanatomist - That is what has been killing me with the 70-200 F4 IS versus the 70-300 F4-5.6. Extra 100 is nice, but I think the 70-200 might be sharper. You didn't like the 2.8 non-IS? I think I read that while it is a little faster a lens, the F4 is a tad sharper.
No difference in sharpness between the 70-200mm f/4L IS and the 70-300mm f/4-5.6L IS. The 70-200mm f/2.8L IS II has a slight edge (very slight) over both.
As awinphoto, the 70-200mm f/2.8L (non-IS) is great from a tripod, or the right choice if you'll mainly be shooting sports/events under dim light and the IS MkII version is not in your budget. The f/4L IS is a tad sharper than the f/2.8L non-IS.
On my copies; the 70-300L is sharper than the already sharp 70-200f2.8 Mk.II at 100mm (The only test I did was at f5.6 @100mm). This is not a competition between the two, but just to let you know how sharp the 70-300L is since you are worried about losing the 70-200 f4.
Attached is a shot I took with the 70-300L; not sharpened, just cropped 100% without resizing. @ 300mm @ f5.6 on an older 5D.
I tested a 70-200f4 once and it seemed sharper than the 70-200f2.8mk.II : I shot them both at their max apertures, i.e. F4 and F2.8 respectively... I know, not fair fight... but told me how sharp the 70-200 IS F4 is wide open.