Not every Mary Sue can have the patience of a saint. After all, if you were that perfect, lesser beings would get on your nerves too. Everyone falling in love with you? It's such a pain! People not always realizing that you are the most kick ass person here and you have to be the one to save the day? Fools! People expecting you to follow the same rules as the other characters? You'll show them! Grr! Get out of Jerk Sue's way!
The basis of this trope is the tendency of many Darker and Edgier writers to create a bitter, ill-spirited, confrontational, or downright mean character and still play them up as an ideal person, or just get away with being a bullying Jerkass. The other characters tolerate the antics (which can range from petty to sociopathic), allowing him/her to walk all over them and talking them up in their conversations with each other. Common synonyms include "strong", "tough", and "rugged" while common topics include how much better they've made their lives through their "Tough Love" or whatever Freudian Excuse supposedly justifies their behavior. It's often an unsuccessful or inept attempt to play on Jerkass Dissonance.
Either way, they're both royal pains in the neck and annoying both to be around and read about. The male version tends to be less common nowadays, (or at least called out more often) as with Marty Stu he's falling victim to men wanting more realistic protagonists.
Compare Draco in Leather Pants, when it's the fanbase itself that overlooks or all too easily excuses a character's jerkish habits, or Creator's Pet, when it's the writers doing it and the fans despise the character as a result. Either of these can overlap with Jerk Sue.
Often overlaps with Karma Houdini.
No examples, please. This only defines the term.