What role does sarcasm play in pride and prejudice? how does its role differ from that of irony? how do both sarcasm and irony affect your understanding of the story?

1- Before being able to answer these questions, you would have to understand the difference between sarcasm and verbal irony. 
Verbal irony happens when a character says something while meaning something different. An example of this would be a man standing in the scorching sun stating "At least the weather is nice today" as he sweats bullets. He doesn't think that the weather is actually nice. Instead, he is pointing out how unpleasant it is by stating what it's not.
On the other hand, sarcasm, while similar to verbal irony, is mostly used to mock and ridicule the recipient of the remark. Let's say that you were to approach this man standing in the sun, and you dared to ask him if he was hot. If he was of the sarcastic kind, he would answer something like "Of course not, Einstein! I'm sweating because I like feeling moist." In a similar remark, not only would he be pointing out the obvious by stating the opposite, he would also be insulting or making fun of you in the meantime.
Now, to your questions.
2- What role does sarcasm play in Pride and Prejudice? 
In both, the book and the movie, it is clear that one of the main things that sets Elizabeth Bennet (protagonist) apart from the other girls is her wit. because of this, she is able to come back with the best comebacks of the Romanticism and Victorian era (Current period in which the book was written and the period that came after). Mr. Darcy (Elizabeth's love interest), who thinks he's all that, is not used to being put to the test but rather having all girls dying to be with him. When he tries to treat Elizabeth as inferior, she demonstrates with her sarcasm that while her socioeconomic status might be inferior, her mind is not. Precisely because of her witty comebacks, Mr. Darcy was able to tell her apart from everyone else and fall for her. Because of this, we can undoubtedly conclude that sarcasm is one of Elizabeth's best features at the center of the story's conflict.
3- How does its role differ from that of irony? While irony is still used in the hands of the characters, this is more importantly used by the author to critique the society she was in. In the time that Jane Austen wrote Pride and Prejudice, going up the social ladder was everything that mattered, and she hated it. She wrote this novel and critiqued that aspect through it, and how women were thought of as useful in terms of marriage. She uses this to present the story's biggest irony, how the characters all preach this lifestyle while actually believing in a better reality. Mr. Darcy had everyone believe that he only cared about his role in society, while in reality being a deeply emotional man. Mr. Bennet, too, wanted his daughters to marry into wealth, but was hesitant to let Elizabeth marry Mr. Darcy for any reason other than love.
4- How do both sarcasm and irony affect your understanding of the story?
While this question is personal, it is easy to see how, without understanding irony and sarcasm, one could misinterpret the characters motivations and true emotions.