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.
Perhaps they are loathed so much not just because they are jerks, but because you are also expected to think they are awesome. They certainly come with most of the other traits of Mary Sueness - being completely untouchable in every way. And of course, despite being a jerk, everybody either completely ignores her appalling attitude, or lets her walk all over them, even the ones who would normally peel off her skin and wear it as a coat for that kind of behaviour. Either way, they're royal pains in the neck and annoying to be around and read about.
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.
Compare and contrast Jerkass Woobie, where a Jerkass is portrayed sympathetically as a flawed, emotionally fragile character. The Jerk Sue can often arise from a failed attempt at this type of character.
Often overlaps with Karma Houdini.
No examples, please. This only defines the term.