Biff and Happy have very different personalities, and this also affects the way each of them treat their father, Willy Loman. Happy's name is very significant-he is a happy-go-lucky character, he ignores unpleasant things in life and just focuses on what is pleasant and what makes him happy. Biff, on the other hand, feels very unsatisfied with his life and blames his father for a lot of the unhappiness.
In the first conversation between Biff and Happy at the beginning of the play, they discuss their father:
HAPPY: You’re not still sour on Dad, are you, Biff?
BIFF: He’s all right, I guess.
We can see here that Happy has a better relationship with Willy than Biff does. They go on to say,
BIFF: Why does Dad mock me all the time?
HAPPY: He’s not mocking you, he...
BIFF: Everything I say there’s a twist of mockery on his face. I can’t get near him.
HAPPY: He just wants you to make good, that’s all.
Happy defends Willy, while Biff criticizes him.
Biff also stands up to his father, while Happy either remains silent or tries to make everyone calm. Biff gets upset when Willy yells at Linda, and Biff says, "(furiously): Stop yelling at her!" Shortly before that, Happy says, "(trying to stop them): Wait a..." Happy just wants everyone to get along.
It is important to note that we learn through flashback about an incident that happened right after Biff graduated high school that changed Biff and Willy's relationship for forever. Willy was away on a trip, and he was staying in a hotel with his mistress, "The Woman," as she is called, and Biff went to find Willy because he needed help. Biff discovered the woman in Willy's hotel room, and after that lost all respect for his father. Before that, he had idolized him. But when he discovers his father is cheating, Biff says to him, "You fake! You phony little fake! You fake!" Happy, however, remains ignorant of this (as far as we know), so he still respects his father.
At the end of the play, in the last scene with the whole Loman family, they fight. Biff confronts his father about the piece of rubber tube that Linda had found that Wily was going to use to kill himself. In this scene, we again see Biff angry and confronting his father, while Happy tries to maintain the peace:
WILLY (sinking into a chair at a table, with full accusation): You’re trying to put a knife in me — don’t think I don’t know what you’re doing!
BIFF: All right, phony! Then let’s lay it on the line. (He whips the rubber tube out of his pocket and puts it on the table.)
HAPPY: You crazy...
LINDA: Biff! (She moves to grab the hose, but Biff holds it down with his hand.)
BIFF: Leave it there! Don’t move it!
WILLY (not looking at it): What is that?
BIFF: You know goddam well what that is.
WILLY (caged, wanting to escape): I never saw that.
BIFF: You saw it. The mice didn’t bring it into the cellar! What is this supposed to do, make a hero out of you? This supposed to make me sorry for you?
WILLY: Never heard of it.
BIFF: There’ll be no pity for you, you hear it? No pity!
WILLY (to Linda): You hear the spite!
BIFF: No, you’re going to hear the truth — what you are and what I am!
LINDA: Stop it!
HAPPY (coming down toward Biff): You cut it now!
BIFF (to Happy): The man don’t know who we are! The man is gonna know! (To Willy) We never told the truth for ten minutes in this house!
HAPPY: We always told the truth!
We see instances throughout this scene of Biff treating Willy in an angry and disrespectful way (although we can understand some of his frustration), and Happy just wants everything to be okay. So the main difference in how the two brothers treat their father is that Happy treats his father as though nothing is wrong, whereas Biff can't take it anymore and feels the need to speak the truth to his father.