To state what the dispute is about and to understand its relevance, one first has to analyse and understand the characters involved, i.e. Thomas Putnam and John Proctor.
In his notes, Arthur Miller says the following about Thomas Putnam:
He was a man with many grievances, ...
... was the eldest son of the richest man in the village.
... he regarded himself as the intellectual superior of most of the people around him.
His vindictive nature was demonstrated long before the witch-craft began.
Thomas Putnam felt that his own name and the honor of his family had been smirched by the village, and he meant to right matters however he could.
Another reason to believe him a deeply embittered man was his attempt to break his father’s will, which left a disproportionate amount to a stepbrother. As with every other public cause in which he tried to force his way, he failed in this. So it is not surprising to find that so many accusations against people are in the handwriting of Thomas Putnam, or that his name is so often found as a witness corroborating the super-natural testimony, or that his daughter led the crying-out at the most opportune junctures of the trials ...
The impression created of Thomas Putnam therefore, is that he wanted to avenge the wrongs he felt that he had been done. The witch trials presented an ideal opportunity for him to do so. Added to this is the fact that Thomas could use his wealth to buy land at auction once a landowner had been arrested and the land was forfeited to the state. He would therefore not only punish his enemies, but also profit from their misfortune.
Arthur Miller says the following about John Proctor:
Proctor was a farmer in his middle thirties, He need not have been a partisan of any faction in the town, but there is evidence to suggest that he had a sharp and biting way with hypocrites. He was the kind of man - powerful of body, even-tempered, and not easily led - who cannot refuse support to partisans with-out drawing their deepest resentment. In Proctor’s presence a fool felt his foolishness instantly - and a Proctor is always marked for calumny therefore.
Proctor, respected and even feared in Salem, ...
What we have here are two powerful characters, one wealthy and the other respected and even feared for his forthright manner and his physical presence. It is obvious that these two men would, at some point or another, clash.
Their first dispute is about the fact that Putnam wants Reverend Parris to investigate signs of witchcraft when the respected Reverend Hale from Beverley arrives. Proctor takes exception to Putnam's instruction, feeling that Putnam is overplaying his hand. He tells him:
You cannot command Mr. Parris. We vote by name in this society, not by acreage.
Proctor clearly resents Mr Putnam's commanding tone. When John criticises the Reverend, Putnam defends the Reverend. Proctor makes a sneering remark about not liking the smell of 'authority' when Putnam says, in support of the Reverend, that there is a faction against the reverend and 'all authority'. The lines of animosity are clearly drawn here.
The argument evolves into a dispute about lumber that John wishes to drag home. Putnam asks him what lumber he is talking about. When John tells him, Putnam claims that the land from which the lumber was taken belongs to him. Proctor tells him that he bought the land from Francis Nurse and Putnam says that that land had been willed to him by his grandfather.
Your grandfather had a habit of willing land that never belonged to him, if I may say it plain.
Giles Corey supports John in this contention, and the two men leave with Putnam hurling threats of legal action and arrest against them.