I used to own the 70-300 and the 55-250, and much preferred the 55-250. I ended up selling the 70-200 and keeping the other lens.
Reasons? In terms of image quality (very good) and focusing speed (fairly slow), the two lenses are virtually identical. The USM on the 70-300 isn't "true" USM. However, IMHO the IS system on the 55-250 is superior - I found it to be much more effective. In addition, my 70-300 wasn't all that sharp over 250 mm, especially wide-open. And I found the additional 15 mm on the wide end of the 55-250 to be more useful than the somewhat soft 50 mm on the long end of the 70-300.
Don't get me wrong; I liked both lenses. However, there's a very significant price difference - the 70-300 is more than twice the price of the 55-250, and IMHO it's very far from being twice the lens - if, indeed, it's better at all.
If I were buying today, I'd have no hesitation in choosing the 55-250. It's a tremendous lens for the price, and performs well wide-open.