AFAIK there are no changes in the frame/chassis itself (although it's possible that the "crash cage" around the electric motor adds some front-end stiffness). I believe, however, that the SE uses the stiffer springs/shocks from the heavier Countryman to help compensate for the additional weight. (This is presumably why the frequency selective damping implemented in all the ICE versions for 2022 is not available on the SE.)
As for the actual handling, I have not driven an SE, but have always found this video interesting: