When Elizabeth expresses to her sister Jane her hesitations at exposing Wickham's character after Darcy confided in her, Jane agrees that they should keep Wickham's history hushed up. However, they disagree on the reasons why. Elizabeth hesitates simply because Darcy has not given her permission to relay a terrible story involving Miss Darcy. Furthermore, she believes that no one would be willing to believe Darcy's account of Wickham, since everyone loves Wickham and dislikes Darcy. Jane's take on the issue, since she is always willing to believe the best about a person's character, is that if they expose Wickham now, they'll forever ruin his reputation. She believes that Wickham may be feeling remorse for having tried to seduce Miss Darcy and is now trying to "re-establish his character," as we see in her lines:
To have his errors made public might ruin him for ever. He is now perhaps sorry for what he has done, and anxious to re-establish a character. We must not make him desperate. (Ch. 40)
Even after Lydia runs off with Wickham unmarried, Jane's opinion does not waver. She still believes that exposing someone's past without knowing his/her present feelings and future intentions is cruel. When Elizabeth grieves over their decision not to expose Wickham, Jane argues, "But to expose the former faults of any person, without knowing what their present feelings were, seemed unjustifiable" (Ch. 47).
However, Darcy disagrees with Jane. While he doesn't say anything in reply to Elizabeth when she bemoans her choice not to expose Wickham's character, we later find out that he actually considers himself to blame for Wickham's current actions, not Elizabeth. We learn that Darcy finds himself to blame in the letter Elizabeth receives from her Aunt Gardiner explaining Darcy's involvement in Lydia and Wickham's forced marriage. Darcy feels that he should have exposed Wickham's character himself in order to prevent any family or any young lady from ever trusting Wickham again and that it was his pride that prevented him from exposing Wickham. Hence, Darcy searched out the couple in London and bribed Wickham to marry Lydia, and his motive was his own self-blame, as we see Mrs. Gardiner explain:
The motive professed was his conviction of its being owing to himself that Wickham's worthlessness had not been so well known as to make it impossible for any young woman of character to love or confide in him. He generously imputed the whole to his mistaken pride. (Ch. 52)
Hence we see that Jane feels that Wickham should not have been exposed so that he can re-establish his character while Darcy feels that Wickham should have been exposed in order to spare any future young ladies Wickham might also try to seduce.