The 70-200 f/4 is a reasonable alternative option. I wouldn't say it is better or worse. per se, but the 70-300 is probably more often a better choice for an amateur (and that's pretty much the only market, as most pros would get a 2.8)
* It is a tiny bit wider aperture, yes, but the 70-300 is f/5 max aperture at 200mm, so it is at most 2/3 stop slower, and rapidly becomes only 1/3 stop slower and then 0 compared to the 70-200 f/4 as you go down in the zoom range. In msot cases this will be trivial.
* It has no IS. This means that for
still subjects, the 70-300 is effectively quite a bit
faster in terms of shutter speed. Although IS doesn't replace slight differences in background blur, nor does it apply for moving subjects.
* It has less reach, and the reach on the 70-300 is still sharp enough to be superior at 300mm to a crop from the 70-200, so that is a real difference.
* The 70-300 has a max magnification of 1:4, the 70-200 f/4 does 1:5, somewhat better macro ability from the 70-300 (it is actually a useful macro lens
* The 70-200 is of course much better build quality, weather sealed and internally focusing and zooming, etc.
* The 70-200 used is still about
twice as expensive though than the 70-300 used
As for sharpness, unfortunately Canon doesn't publish MTF charts for both at 200mm for a fair comparison, and DxO Mark hasn't tested the non-IS 70-200 f/4, which is the comparably priced lens to the 70-300 IS USM. But here is a comparison with the 70-200 f/4 IS. Reviews suggest that they are similar in sharpness (IS and non IS):
Acutance is maybe 5% better for the 70-200 @200 in the center and 25% better at the edge at a fair comparison of f/8 for each.
In my experience, when you are shooting still subjects, though in normal lighting?
The lack of image stabilization will often cause more pixel-level blur than the sharpness will give back! This is just from me using the 70-300 with and without IS, but the principle obviously applies to any other comparable lens. These charts are tested on a rock solid tripod, not hand held, remember.
If you're shooting a still subject at 1/2000th of a second? That's a non issue. But how often do you do that? More often you shoot at 1/400th or similar, and this becomes very much true.
So I would buy the 70-300 if:
* You want to do macro stuff
* You want to shoot handheld still subjects
* You need all the reach you can get like for small wildlife
* $250 matters to you.
I would get the 70-200 f/4 if:
* You will be in adverse environmental conditions a lot like rock climbing with it or lots of rain
* You will be shooting on a tripod OR shooting fast moving subjects.
* You will be shooting video with it (parfocal / constant aperture is important)
* You don't have a strong need for 300mm.
In most cases, I think the typical person would fit the 70-300 better. People doing extreme sports with DSLRs on their hip are not very common, nor are casual videographers using 200mm lenses.
The most common situation where the 70-200 is a better choice is for people shooting moving subjects like sports.