In Act 1 of The Crucible, why do Putnam and Proctor argue?

Thomas Putnam takes Reverend Parris's side in the minister's argument with John Proctor. Parris feels that Proctor is the leader of a faction that is trying to remove him from authority, and Putnam alleges that Proctor, among others, is actively working "Against [Parris] and all authority!" Then, when Proctor claims that he has some lumber to move, Putnam wants to know "What lumber is that you're draggin' [home], if I may ask you?" Proctor says that it is wood from his land near the river, and Putnam grows angry. He says,

Why, we are surely gone wild this year. What anarchy is this? That tract is in my bounds, it's in my bounds, Mr. Proctor.

However, Proctor claims that he bought the piece of land from Rebecca Nurse's husband, Francis, several months prior. Putnam, then, claims that Nurse "had no right to sell it" because his own grandfather clearly left it to him in his will. Proctor then argues that Putnam's "grandfather had a habit of willing land that never belonged to him [...]." In other words, Proctor alleges that Putnam's grandfather was not particularly honest or forthright. Giles Corey agrees that Putnam's grandfather did make a habit of this and "nearly willed away [Giles's] north pasture" as well. Putnam then threatens Proctor, saying that he'll "fight to drag it home" if he tries to move one oak that belongs to him; Giles accepts the challenge, insisting that he and John would win in a fight against Putnam.

In addition, there is bad blood between the Putnams and the Nurses, and the Proctors and Coreys are close with the Nurses, so this creates hard feelings between them and the Putnams, as well. Miller tells us that Putnam had been engaged in a "land war" with Francis Nurse, and the Nurses apparently blocked Putnam's candidate for the Salem ministry as well. We can see Putnam's greed, too, in the apparent land-grab he makes when his daughter accuses his neighbor—a man whose land he's been wanting to buy—of witchcraft.

What is the dispute between John Proctor and Thomas Putnam in Act 2 of The Crucible?

There is no dispute between John Proctor and Thomas Putnam in act 2. Although John Proctor does not like Thomas Putnam and views him as an arrogant, greedy man, the two men do not exchange words in act 2 and Thomas Putnam does not even enter the scene. However, a dispute between John Proctor and Thomas Putnam does take place in act 1. When Thomas Putman demands that Reverend Parris call Reverend Hale from Beverly to search the town for witchcraft, Proctor intervenes and tells him:

You cannot command Mr. Parris. We vote by name in this society, not by acreage. (Miller, 28)

Thomas Putnam responds by bringing up John Proctor's less-than-stellar church attendance record. In addition to Proctor arguing with Thomas Putnam for attempting to use his influence over Reverend Parris, the two men also argue over lumber and territory. When Proctor asks Giles to help him haul his lumber home, Putnam accuses Proctor of stealing his lumber from his tract of land. Although Proctor mentions that he bought it from Goody Nurse, Putnam insists that the property belonged to his grandfather, which means it is now his.