Jane Austen carefully establishes equality between the two characters but not always similarity. Their positive qualities largely complement each other.
One quality they share in equal amounts is (unsurprisingly) pride. In Lizzie's case, she has a strong ego and is defensive about her family's position and reputation, not just their limited means. Lizzie would like to indulge her own interests in books as well as socializing, and is proud of her own learning and intelligence. This aspect of her pride sometimes causes her to seem cruel or at least thoughtless, as for example concerning Charlotte's marriage.
Darcy's pride stems from his social position, as he tends to place credence in rank. More than that, however, he has confidence in his own qualities as a leader and protector. While this confidence seems over-inflated, we later learn how he won it through his efforts to care for his little sister, including rescuing her from Wickham.
Another important area of equality is intelligence. Their initial encounter features clever banter, showing their verbal abilities are well matched. It takes longer for them to realize that each other's brain power goes deeper and that both are similarly serious people, as Lizzie in particular enjoys joking around. The ways each person grows are different, but Austen shows each as maturing in a way consistent with their nature, as well as growing toward each other, thereby helping the reader believe they will remain well-matched partners throughout their marriage.