Mr. Armond is a New York City lawyer who spends the spring and summer on his farm, which borders on the Breen property. He's very well-off, and his property has several houses and a well-manicured lawn, and he's got all the newest gadgets in his barn: a wagonette, a buckboard, a buggy with red wheels, a cutter, and a surrey with fringe on top to boot. Basically his stash is like the old-timey version of having all the shiniest new sports cars.
All this is enough to make Tom feel "mighty insignificant" (34.25). Polly Ann knows the feeling: Mr. Armond is her nemesis. She doesn't actually have anything against the man himself, but rather what he reminds her of, which is having to peddle berries at the back door of his fancy house for her own no-good father. It's a matter of pride for her.
When Tom pulls up in his old horse and wagon to make an offer on the barn after Mr. Armond buys the Breen land, there is the sense that he and Mr. Armond are not exactly on equal footing. For one thing, Mr. Armond almost literally says, "get off my lawn," like some crotchety old man guarding his trash cans. (Okay, he actually says "I wouldn't want your horse wandering over the lawn, you know. He would leave tracks in the grass" [34.22]. But still.)
For another, Tom takes his hat off out of respect, but Mr. Armond does not. Nevertheless, Mr. Armond is perfectly amiable to doing business with Tom, and he even knocks $25 off Tom's offer price of $100. But so he doesn't appear too "charitable" (35.22), he stipulates that Tom also has to demolish the old Breen house. Tom puts $50 down, and they agree that he will pay the remaining $25 when Tom's work on the property is done.
By the time Tom returns to pay the $25, life for the Dolans has changed big time. Polly Ann rides up with Tom in their new wagon with their new team, and she is dressed in her best outfit. She's now the kind of woman Mr. Armond invites in for tea with his wife (Polly Ann politely declines) rather than the kind of woman begging for money at the back door of the home. That, plus the compliments Mr. Armond gives her on her fine son, makes for vindication to the max for Polly Ann.