Simply put, a Mary Sue is a character who lacks substantial flaws. When I say flaws, I mean traits that emotionally, morally and physically limit the character. Take spider man for example. He's not a Gary Stu, because he has both physical limits (while he can take a punch, he's not invulnerable), moral limits (he's a good-hearted character, but displayed a desire for revenge in the movies) and emotional limits (suffers chronic heartbreak and endured emotionally hard things) Early superman had little to none of these traits. He was a Gary Stu. Now Mary Sues are not well written characters, but if the rest of the story they are in is good most people wont mind. That's why characters like Superman and Goku are still so popular despite being rather bland. It all depends on the story itself.
Anyway, I personally think that Twilight could use more moral limitations but I'm not gonna start that argument again....