I guess the other part of this is for Killswitch - you're also asking about two almost completely different lenses on an APS-C. The 70-300 is a 114-480 or so equivalent. The 55-250 will remain a 55-250.It does indeed remain 55-250, but that's 88-400mm "equivalent". The 55-250mm spec is the true focal length range, not the full frame equivalent focal length.
I thought we were talking about APS-C... Hence why I said it is a 55-250 on APS-C and a 70-300 is a 114-480 on APS-C, via the actual field of view available, never stated anything about a FF equivalent focal . They are still, in my opinion, two completely different lenses on an APS-C sensor. Unless he's contemplating porting an EF-S 55-250 to an FF body, which would be beyond my reasoning for this lens.
We are talking about APS-C, but so what? I suspect you are suffering from a (rather common) misconception, that EF-S lenses have a focal length that is somehow 'adjusted' for an APS-C sensor. That is not the case. Focal length, by definition, is the physical distance from the rear nodal point of a lens to the image plane. In other words, a lens marked 55-250mm has a focal length of 55-250mm, and a lens marked 70-300mm has a focal length of 70-300mm (ignoring focus breathing) - regardless of the size of the sensor behind that lens, or even whether there is a camera there at all. Focal length is focal length - it's an intrinsic property of a lens. Canon made a lens with focal lengths of 55-250mm, they didn't make a lens with focal lengths of 34-156mm then lie and print 55-250mm on the barrel...
Yes, an EF-S lens will only mount on an APS-C sensor, but that doesn't affect the focal length. If you want to compare fields of view, either use no correction or 1.6x, but you have to apply it to both lenses. So, you can say 55-250mm vs. 70-300mm, or if you want to compare the full frame equivalents, as elflord stated, a 55-250mm on APS-C will produce a field of view equivalent to a hypothetical 88-400mm lens on FF, just as a 70-300mm on APS-C is equivalent to 112-480mm on FF. To repeat, you apply the crop factor to all lenses, not just to EF lenses mounted on APS-C bodies.
Are they different? Yes - by 24mm on the wide end and 80mm on the long end, but not, as you seem to be suggesting, by more than that (e.g. you mentioned the difference between 55mm and 114mm, but no, it's between 88mm and 114mm, and 400mm to 480mm on the other end, not 250mm to 480mm). Different - but not so different after all, in terms of focal length, at least.