Coming from a Michelin CrossClimate 2, the thing I had noticed IMMEDIATELY is how much more comfort and less noise these tires offered.
The bumps went from hurting my back to not being felt, and the noise was less pitchy.
The car was as planted in the dry, and responded faster and better to turns. Granted, CC2 was only slightly more planted in the rain, but ASC2 gives you the perfect amount of confidence in the car. Only slightly tried in the snow, and was perfect. But no matter the temperature around, the tire remained lovely, balanced, and linear (predictable).
Weelspin went away (where I had more occasion with CC2, especially in harsh ascends).
The breaking is more linear than the CC2: Whereas the CC2 would "bite" the road immediately, ASC2 is a more linear stop. I find it to be a bit more comfortable way, especially in the city.
And finally, plenty of rim protection in ASC2, something inexistent on CC2.
Overall, really good tire that I will be buying again (I rarely need the summer/winter split) and would absolutely recommend to anyone.