Heat-treatment raises the yield & ultimate-strength of the tubing. These are the load-points where it takes a permanent set (bent) or breaks completely. Heat-treatment however, doesn't change the modulus or stiffness of the material, it will still deflect the same amount given the same load. It will be able to take a higher load and deflect more before permanent damage

As far as those qualities listed and the differences between 853 and 631, I doubt you'd notice the any difference at all. Fully-optimized frames from each one might result in a slight weight-savings, I'd say about 1/4 lb max.