Hi,
A good copy of Canon's 55-250 should be as sharp as a 70-300. The 70-300 vignettes less. The 55-250 is at best at f8 all the way. The 70-300 is good even wide open up to about 180mm. Beyond that it needs to be stopped down for sharpness, at 300mm even towards f11 to get really sharp. Not much difference in focus speed between the two, but I've experienced some focus inconsistancy with the 55-250 (on a 40D and sometimes it would not even focus at all with a 350D at longer focal length).
Since the range from 250 to 300mm on the 70-300 is not it's best part I see no real advantage of it over the 55-250, except for focus with older DSLRs.
A 70-200 f4 non-IS is not thát much more expensive than a 70-300. It has a much faster full time USM, is constant f4, has superior build and optical quality. No IS though, but maybe something to consider.
If you go for a 55-250 be sure to get a good copy. Both my father and brother needed two copies to get a good one. The one my dad has now is really good.
Regards,
Sandor.