Well, the 70-300 is pretty much just over twice the price.... both have IS so its not that and neither are white.
With all those things taken into consideration, I'd say the 70-300 must have something else about it to warrant the price difference.
390g vs 630g the weight probably starts to give you an idea where the differences lie (ie extra elements). The 55-250 is EFS vs the EF fit for the 70-300 (remembering an EFS mount is only for Canon APS-C cameras), which could explain some of the weight difference, but the number of elements is definitely the real cause.
Lastly, the 55-250 is only micro motor drive on the AF, whereas the 70-300 is USM. The latter is both faster focusing and quieter.
The 70-300 is a very good lens though actually, the only thing better in that rough zoom range is the Canon 70-200 family, although once again you are starting at a lot more money.
You can pick up the 70-300 for just over 300 quid including shipping from HK, which is an excellent price for a good all round performer.