The fundamental difference between Hal and Hotspur emerges when Hotspur tries to rally his troops. The speech is a disaster, with Hotspur even admitting his inability to speak, "That I, that have not well the gift of tongue" (5.2.77). Not only are his words poorly chosen, but he is interrupted twice by messengers, making the speech a complete disaster with news of King Henry's imminent arrival. Hal would never have had this problem because he has a way with words and is able to command attention. I believe that is why Hal won in the end because he can be a ruler whereas Hotspur may be good at war could never rule as king.
The relationship between Falstaff and Hal comes to a breaking point while on the battlefield. Hal orders Falstaff to draw his sword and fight, an order which Falstaff refuses to obey. He instead pulls out a bottle of sack, at which Hal cries out, "What, is it a time to jest and dally now?" (5.3.54). The inability of Falstaff to realize the serious of the situation is marked by Hal throwing the bottle at his former friend. Which is why Hal is making extreme progress at becoming King because he is cutting ties with the friends that are holding him back and are a bad influence on him, such as Falstaff. All this shows is that Hal is ready to be King.
Hal's pardon of Douglas at the end of the play marks a shift in the way he will rule as king versus how Henry rules. Henry orders Vernon and Worcester to be put to death for their crimes, Hal instead chooses to take a risk and pardon his enemy. This relates to the fact that Hal is a redeemer of England, a role he can only play by creating peace. In pardoning Douglas, Hal shows not only great statesmanship, but also confidence in his own abilities to win Douglas' support in the future. And this reiterates that Hal will make a better king than Henry and can handle uniting England instead of splitting it like Henry seems to be doing. That is probably why Hal will become a legend whereas Henry will be forgot.