With this comparison on a Canon EOS 7D, we see a small advantage for the 
Canon EF-S 55-250mm f/4-5.6 IS II  with a DxOMark score of 8, 2 points more than its predecessor. This  difference doesn’t come from the resolution. With a score of 42lp/mm and  44 lp/mm, for the 2 lenses, the sharpness didn’t improve. The  distortion and vignetting are also identical.
But, the  transmission of the new version is on average 1/3 stop brighter. And,  the chromatic aberrations are better controlled in the corners at 55 mm.
Conclusion
Very little improvements, and no regressions, so overall Canon maintains the quality of its kit telephoto zoom.
In complement with the 
Canon EF-S 55-250mm IS II, this lens will enable amateur photographers to be prepared for most situations