I hear the 70-300L is as sharp as the 70-200 f/4L IS, which is very, very, very sharp for a zoom lens.
The 28-300mm is nice because of all the range in one lens, but the image quality suffers a bit. If the maximum image quality is your first priority, having two lenses cover that range (like briansquibbs suggestion) would be the best way to go.
If you prefer the versatility of having all range in just one lens, the 70-300L is a fine lens. Its image quality is not the best of Canons L lens range, but understandibly so - it's amazing how they got this much quality in a >10x zoom to begin with.
Another good consideration would be to become acquainted with the push-pull zoom mechanism that the 28-300L has, and not the 70-300L (traditional 'rotating ring' zoom). Some people hate it, but some people really like it. If you turn out to be one of those people that REALLY likes that, it'd definately be something to consider. And vice versa, if you try push-pull for a few weeks or so but really can't get used to it, then the 70-300L is your best bet.
Further notable differences (apart from the image quality) are that the 28-300 is about 60% heavier than the 70-300 (1.7kg or 3.7lbs versus 1.1kg or 2.3lbs), and that the 28-300 has an older generation IS with 3 stops improvement, vs 4 stops for the 70-300L (which is a very new lens).