In The Kite Runner, Hasaan used his slingshot against Assef. 26 years later, Sohrab also used the slingshot to injure Assef. What does each action...

The similarities between both of these instances are at least as striking as the differences. In each case, Amir is rescued from Assef. The socially powerless character (Hassan/Sohrab) protects the socially more powerful character in an act that dangerously transgresses social boundaries. 

In each case, Amir is the one who, according to ethnic hierarchies in place, should be the person to take charge of the situation. This means that for Hassan and Sohrab the act of using the slingshot (or threatening to use it) functions as a rebellion against the social structure. Crucially, this rebellion is performed as a moral action. Hassan and Sohrab are able to do the right thing, despite the likely consequences. They may be socially powerless, but they are morally powerful. 

This is one of the meanings behind each act that does not change from episode to episode. What does change is Amir's character. When Hassan saves Amir, Amir finds himself unable to reciprocate when Hassan is in need.

"I had one last chance to make a decision. One final opportunity to decide who I was going to be. I could step into that alley, stand up for Hassan—the way he’d stepped up for me all those times in the past—and accept whatever would happen to me. Or I could run.

In the end, I ran."

However, after years of running away from his shame for failing Hassan (then betraying him as well), Amir comes back and tries to save Hassan's son. 

Amir has grown into some strength, morally speaking, though he is still unable to defeat the monstrous figure of Assef alone. Thus in the first slingshot episode, Hassan acts as the lone moral agent, demonstrating an inner strength that is not supplemented or balanced by a moral power in Amir. The result is a break in the friendship and brotherhood of Hassan and Amir that takes place on dishonest and dishonorable terms.

In the second slingshot episode, Sohrab's courage is supplemented by Amir's courage (or, vice versa, Amir's courage is supplemented by Sohrab's courage). The result is a union of the family on honest terms.