While dramatic irony is a classification of situational irony, along with tragic irony, and Socratic irony, the difference is that in dramatic irony the words of characters come back to haunt them, whereas situational irony involves a turn of events in the plot.
In Pride and Prejudice, an example of dramatic irony can be seen when Elizabeth begins to reproach herself after reading Darcy's letter. When Darcy proposes to her, she lays out many accusations about his treatment of her sister and his treatment of Wickham, concluding that he is a prideful and detestable man. After reading his letter, she is forced to eat her words and realizes that all of her accusations stemmed from prejudicial judgement of his character. She calls the realization a "humiliating discovery" that she considers "just" because she had been blinded by Wickham's story and blinded by what she perceived to be Darcy's pride (Ch. 13, Vol. 2).
An example of situational irony would be Lady Catherine de Bourgh's attempted interference in Elizabeth's engagement to Darcy. Lady Catherine condescends to visit Longbourn with the sole desire of telling Elizabeth that she is not worthy of marrying Darcy and demanding that she promise not to accept his marriage proposal. The irony is that Lady Catherine makes this visit just when Elizabeth is wondering what Darcy thinks of her, believing that he no longer cares for her (Ch. 12, Vol. 3). The further irony is that Lady Catherine's visit, contrary to Lady Catherine's desires, leads Elizabeth to hope, even just a little, that Darcy still does care, because Lady Catherine would not have condescended to visit if she had not genuinely heard that Darcy meant to propose (Ch. 15, Vol. 3). Yet there is even more irony in this situation because Elizabeth's refusal to promise Lady Catherine never to accept Darcy's proposal actually gave him his first instance of hope that Elizabeth now cares for him because if Elizabeth still detested him, Darcy knew that she would not hesitate to say so, even to Lady Catherine. As Darcy states, "It taught me to hope...as I had scarcely ever allowed myself to hope before" (Ch. 16, Vol. 3). Hence, in two respects, Lady Catherine's visit did exactly the opposite of what she intended to do.