In what ways is the relationship between Lydia and Mr. Wickham similar to and different from that between Elizabeth and Mr. Darcy?
The first part of your question is thought-provoking! The only similarity that I see between these relationships from Pride and Prejudice is that Mr. Bennet is shocked by both and Mrs. Bennet is pleased by both (eventually!). Lydia's relationship/elopement with Wickham is surprising to her father (although Lizzy predicted that something similar would happen), and Elizabeth's relationship floors her father because he thought she disdained Darcy. The girls' mother is, of course, thrilled because she thinks that she has now "matched" three of her daughters.
Differences between the two relationships abound. First, Wickham is everything that Darcy abhors, and Darcy would never treat a woman in the manner in which Wickham treats Lydia. Secondly, Darcy and Elizabeth's relationship is founded upon well-matched intelligence, wit, and eventually complete honesty. Lizzy's younger sister Lydia, unfortunately, does not enjoy the same attributes in her marriage. In fact, she is too silly to even realize the predicament that she created for her family. Wickham is a manipulator, especially of young girls, and will most likely never reveal to Lydia the true cause of the friction between Darcy and himself. Finally, Elizabeth and Darcy are bound to live a life of luxury and happiness, while Wickham will always be dragging Lydia behind him as he searches for his next money-making scheme. If Lydia does ever mature, she will certainly be unhappy to be married to such a fop.