EDIT : The 70-300 is an IS only model... The lower cost lens in that range is the EF 75-300 f/4-5.6 . There is another EF 70-300 f/4-5.6 L USM, which more than twice the cost of the non L version. And the correcting element used for IS would make a difference in optics as well. Thanks @Michael Clark for correcting me!*
Depends on how you plan to use the camera... if you feel that you're going to use the lens mostly in ample light and possibly for most of the time mounted on a tripod... then the 70-300mm is a nice choice if you want the extra zoom.
The 55-250mm has Image Stabilization so is easier to use hand-held, and so is better when hand held at lower shutter speeds. I also believe it is a newer design, so you might get slightly better optics. It is an EF-S lens, so wont mount on a full frame camera. Keep that in mind if you ever consider upgrading to a full frame.
The 70-300 has another model that comes with Image Stabilization, but is significantly more expensive, without any change in the optics itself.
Having used both lenses, on a crop frame body, I find that I do prefer the 55-250, it just is a better all round lens. But you should evaluate by your own specific needs.