One of the most important characteristics of the relationship between Troy and his son Cory is that they are effectively repeating history (and, ultimately, breaking the cycle of history).
As a young man, Troy gets into a fight with his own father over a girl. In this fight, Troy stands up to his father and is kicked out of the house. This begins a life of crime for Troy that lands him in prison.
As a father, Troy plays a big part in generating a conflict with his son that also causes a break between them. Cory leaves home and is prepared to never forgive his father and never speak to him again. However, when Troy dies and Cory returns home, Cory is persuaded to forgive his father, at least tacitly, by agreeing to attend his funeral.
The cycle of conflict between father and son is historic, as this synopsis suggests, and there is a danger that it may continue to be perpetuated. Cory's decision to yield along with his choice to enter the army (instead of leading a wild life) also represents a change and shows that Cory is a different man that his father and grandfather before him.
Taking a more detailed look at the relationship between Troy and Cory, we can note that Cory feels Troy may not love him -- and certainly does not like him.
He asks his father in Act I, "How come you ain't never liked me?"
"Liked you? Who the hell say I got to like you? What law is there say I got to like you?"
He continues on by saying that the only rule he feels compelled to follow in regards to Cory is one of duty. It is his duty to clothe, feed and protect his family. As he says:
"It's my job. It's my responsibility! You understand that? A man got to take care of his family. You live in my house...sleep you behind in my bedclothes...fill you belly up with food...cause you my son. You my flesh and blood. Not 'cause I like you! Cause it's my duty to take care of you."
Troy's sense of duty is strong and, throughout the play, remains undeniable and consistent. His sense of loyalty is lacking, however, as he cheats on his wife and proves willing to lie to his employers.
Cory feels Troy's lack of affection keenly and wishes that his father would yield. This term, yielding, may be the key term in the father-son relationship. Neither Troy nor Cory is initially willing to give in, although Cory is the more sentimental of the two. Regardless of his youth and sentimentality, Cory refuses to quit the football team as his father demands.
Cory's stubbornness shows that he is very much like his father, a strong-headed and confident person. Only in the end, when he agrees to go to Troy's funeral, does Cory truly distinguish himself from his father.
One more point seems critical in a discussion of Troy and Cory Maxson. Troy harbors an undying but quite bitter (and failed) dream of becoming a professional baseball player. He was a good player, but too old for the major leagues by the time he got out of prison. Troy refuses to admit he was too old, however, and prefers to blame race politics (e.g. racism) for his failure to make the Big Leagues.
While there is a grain of truth to Troy's complaint and ample reason for him to point out the racist policies that might hamper him elsewhere in his professional and social life, Troy's dream of sports success was ruined by his own faults and flaws -- by his life as a criminal.
In this way, Troy's relationship to sports is somewhat complicated and less than pure/innocent. It is tainted. Cory's ambitions with sports are equally complicated but not at all tainted. Cory wants to use sports to get a scholarship and go to college. Thus, Cory brings his own motives to his relation to sports.
Due to the complexity of Troy's views of sports, we can only ask if Troy demands that Cory quit the football team as a means to protect his son from disappointment and hurt; from becoming a subject of racism. And we can only ask if he is instead engaging in a competition with his son, setting a limit on Cory so that Cory will not surpass his own achievements. Either of these possibilities is possible.
In any event, Troy refuses to understand Cory's very specific ambition to get into college on a scholarship and so, by default, sets a hard limit on the possibilities of Cory's life. When Cory enlists, he is choosing perhaps the only option left open to him to maintain an expansive view of the life ahead of him. This trajectory is a crucial difference between Cory and Troy as Troy notably spends his time railing against the limits (fences) that have hampered and hindered him from becoming what he might have been in his life.
Cory still has a chance to hit a home-run, as it were, with his life.
To encapsulate the relationship between Cory and Troy, we might describe the dynamic between them as competitive and rooted in history.