A man is king of his castle, as they say, and Pemberley reflects Darcy's true character. Everywhere Elizabeth looks, she is impressed by Darcy's good taste. She contrasts the estate with Rosings, where much of the furnishings and décor are deliberately ostentatious. So while Lady Catherine and Darcy may come across as similar – both haughty, cold, and proud – their essential characters are quite different.
In a novel where the spoken word rules the day, and where private thoughts don't have too much presence on the page, letters are basically a stand-in for the interior lives of the characters. Usually the letter doesn't reveal anything that can't be figured out from the way the letter-writer speaks. Instead, what happens is that it's the letter readers that get to react in a revealing way – since they aren't responding to someone's face, but are instead dealing with the words in relative privacy, or at least behind the person's back. For example, the letters Mr. Collins and Lydia send to the Bennets each get a family-wide reaction that is all the more honest because the letter-writers aren't there in person.