I chose between these two lenses a couple of years ago and went with the 70-300. Been extremely happy with it.
The reviews on Fred Miranda indicated the 300 was an excellent lens and the sharpness could rival the L except at the long end. In my own experience it does get soft past around 250, so I don't use it past there. Also, you can shoot at about F5 at around 250 so that is a little benefit as well.
It's actually the IS that has turned out to be the biggest benefit. I bought the lens pretty much solely for optics and saw the IS as just an extra something that won't be hugely useful, but it is a huge benefit. I can shoot at very low speeds (1/60 at any focal length is zero problem, and handheld I can easily go to 1/20, 1/10, even 1/5, especially if I have something to lean against. Also, being a telephoto I'm sometimes shooting through other objects such as foliage, and since I use the center AF point the IS allows me to get get super steady and hence extremely precise and fast with what I want to focus on. With a normal telephoto, the movement of my hands means the AF point can easily land over something else like a leaf. With the IS I can shoot through very dense foliage easily and nail the AF point over the subject.
Downsides compared to the 70-200 are that barrel length obviously changes as you zoom in, and the front element rotates so will mess with your circular polariser position. I don't find it a problem though, because the circ-p is usually the very last thing I adjust. And of course, the focus and zoom rings don't feel as nice as an L, and it's not constant F4. But the little extra focal length, IS, and price are the big upsides.
I'd be happy to snap some examples and post crops so you can see how the sharpness is at different focal lengths.
