Haemon's arguments to Creon do not fundamentally change over the course of their interaction. What does change is the bond between son and father.
Haemon enters claiming to be loyal to his father's orders. He does this to placate the king, who greets him as he enters as arriving to "fight your father for your bride." Haemon argues that mercy should be present for both Creon's reputation as a ruler and given the fact that the city is sympathetic to Antigone's plight. Haemon weaves both parts of his argument in a tapestry of a son's love for his father:
Even though I'm young, a good idea might(730)
come from me: It would be best by far
that man be born full of all the knowledge
there is, but, if it usually happens
not to turn out that way, to learn from those
who speak well is a good substitute.
From this, the exchange between father and son becomes terse and very pointed, with Creon suggesting that his son has little idea of what he is discussing. At the same time, Haemon's emotions become evident. The use of "stubborn" in the son's description of his father is noteworthy. This helps to reflect how there might be a genuine disagreement of policy as well as a generational rift between father and son. Creon furthers this rift by denigrating both Haemon and Antigone in suggesting that his son should not be so loyal to "a woman."
The conclusion of this is a severing of bonds between father and son:
She'll not die beside me, and you will
never lay your eyes upon my face again,
so rage with any of your friends who can bear it.
In this, the fundamental argument between both do not change. Yet, the relationship between them is forever altered, as Creon will never see his son alive again.