Well, it essentially has to do with which of the two is more willing to do whatever it takes to win.

“You think you can make me stop the order? You think you can make me do that?”

“Yes. And so do you.”

“Sherlock, your big brother, and all the King’s men couldn’t make me do a thing I didn’t want to.”

“Yes, but I’m not my brother. I am you. Prepared to do anything. Prepared to burn. Prepared to what ordinary people wouldn’t do. You want me to shake hands with you in Hell? I shall not disappoint you.”

Sherlock is convincing Moriarty that he is willing to do anything to win (AKA stop his friends from being killed).

Moriarty doesn’t believe him and calls him “ordinary” and “on the side of the angels.”

Sherlock refuses that sentiment, claiming that he “may be on the side of the angels, but don’t think… that [he] is one of them.”

Moriarty seems to analyze him for a moment after that is said, and he apparently changes his mind. He says that Sherlock isn’t ordinary, but he is instead just like him. 

To me, it seems like Moriarty is trying hard to get Sherlock to admit to that, and he is exceptionally pleased when he does say that he is not an angel.

Upon agreeing, Moriarty realizes that Sherlock could almost definitely get the recall code out of him, and he knows that to win, he must prevent that from happening.

“As long as I’m alive, you can save your friends. You’ve got a way out. Well, good luck with that.”

When Moriarty is dead, Sherlock is left with no other option; he knows he has to jump. He knows he has to go to the same extent as Moriarty did to win.

I hope that helps… At least somewhat.