I have the 70-300, never had the 55-250, but I have been very impressed with the quality if there is enough light to stop it down a bit. I actually bought it after I got my 70-200 2.8IS in an effort to have a lighter travel kit (55-250 wasn't an option then). Stop the 70-300 down and it's as sharp as my 70-200 and almost as nice color/contrast. The 70-300 really is a hidden L lens if you have enough light. Problem is, you can't just create light- unless you have a faster lens. And full telephoto at f8?? You better have a lot of light if you are shooting anything with a heartbeat. This is why people often suggest the 70-200 (which doesn't seem a competitor at all) in lieu of either the 55-250 or 70-300.
I've recently considered selling both my 70-200 2.8IS and 70-300 in exchange for a 70-200 f4IS now that with my 7d, I can crop like crazy and still have large enough files to print at 8x10. I think even 100mm shorter, it's a far better option if you are planning on shooting at full reach a lot.