Note that in your construct so also shows degree, extent:
to an indicated or suggested extent or degree (M-W)
In
The kid was so brave as to amaze all his classmates.
the amazement of the classmates is in fact the result/effect of the degree of the kid's bravery.
There is negligible difference between your example (which is more formal) and the variant with that
The kid was so brave that all his classmates were amazed.
There are contexts, however, when the two constructs do differ. Grammarphobia explains
The difference is clearer when the consequence or result is more stark, as in
(1) The dose was so large as to be fatal.
versus
(2) The dose was so large that it was fatal.
#2 is the stronger statement. Why is this? Because the “so … as” constructions indicate extent or degree, while the “so … that” constructions indicate an actual consequence—in other words, a theoretical versus a real result.
To put it more clearly, (1) says that the dose was so large that it was able to bring/could have brought death, whereas (2) makes it real, saying that the dose was so large that it provoked someone's death.