By receiving a letter, JA makes Elizabeth work out the truth for herself, by herself. She is able to read the letter over and over, and gradually put together, from her own knowledge and the information in the letter, the real truth. If Darcy had simply told her 'Wickham is a liar' she would not have believed him, being at that point biased in W's favour. But reading the letter, and weighing their two claims, she then recalls Wickham's former claim that he would never disgrace Darcy - which he of course tries to do in Meryton; and that he would stand his ground, when he actually leaves Meryton the evening of the ball. Her own information on Bingley and Georgiana, which she obtained from C Fitz., adds to verify the letter. As a strong character who values her own discernment, Lizzy needs to work it out for herself, convince herself, for her to fully accept the truth. Using a letter rather than a conversation enables her to do this.

It has also struck me that Darcy writing a letter to a young woman he is not engaged to and who is not a family member would perhaps be a slight breach of decorum ?? In S&S Elinor says she would accept correspondence between her sister and W'by as proof of engagement. Not that this is much of a 'correspondence' between Lizzy and Darcy, but it has seemed to me he is taking a risk - by giving Lizzy written proof of his sister's near elopement, by suggesting she approach C Fitz. for verification on very personal and private matters, by writing to a woman he has just proposed to. Once Lizzy has come to see that he is innocent as far as poor treatment of Wickham is concerned, even if she had wanted to she is not in the position to respond to his letter by letter, as this would be approaching an open correspondence - quite shocking :) .