A "Mary-Sue story" can be described as a wish-fulfilment fantasy on the part of the author, in which a thinly disguised avatar of the author is placed into the story, a character too good to be true. Some authors of Mary-Sue stories therefore defend themselves with a cry of "MariSu doesn't resemble me at all!". However, since a variation on the Mary-Sue is when a canon character is made into someone too good to be true, (these are given names such as Willow-Sue or Mulder-Sue etc) then one can see that the emphasis of Mary-Sue's critics is not so much on the self-insertion as on the wish-fulfilment. (That's why I like to use a separate term, "self-insertion", for stories in which the Author appears as a character β because while these can often be Mary-Sue stories as well, it is actually possible to write self-insertion stories that aren't Mary-Sues, and I'd rather have a separate term than confuse the issue by calling such things "good Mary-Sue stories".)
Mary-Sue and Romance
There are some fandoms where, as soon as an original female character appears, she is accused of being a Mary-Sue. Others restrain the accusation until the female in question is set up as a romantic interest of one of Our Heroes. Why is the question of romance such a sensitive one? Is it that most romances are badly written, or is there a significant difference between the introduction of an original character as opposed to a romance between existing canon characters?
I think the last is probably an important factor. Part of it is plausibility. If a character is already there, then the Hero in question will have a better chance of falling in love with her, simply because they're already interacting and getting to know each other. Secondly, a canon character (whether regular or recurring) already has a reason for being there, a reason other than being a love-interest. They already have a character of their own (even if it's very sketchy) and therefore have a better chance of being of interest for their own sake.
An author who wants to introduce an original female character for the purposes of romance has a number of hurdles to overcome, temptations to avoid. If her sole, or main purpose in being in the story is as a romantic object for Our Hero, then the temptation will be to try to prove to the reader that she is "good enough" for Our Hero, and from there it's a small step to "too perfect" and Mary-Suism. Even if the author isn't trying to prove anything, there may still be a Babe-of-the-Week effect, making the character a love-object and not a person.
Maybe it's just personal taste here, but I've always found romances more interesting if they developed over a longer period rather than quickly. Which, for fanfic, could mean over the course of more than one story. Which means the lady in question needs to have some other purpose to enable her to hang around with Our Hero.
Mary-Sue and Perfection
Whether or not a Mary-Sue is romantically involved, one of her most obvious characteristics is that she is too good to be true, unbelievably perfect, and often having other signs of "specialness", such as an exotic name. However, it isn't so much the absolute signs of her amazing good traits, so much as how much better she is than those around her. It's not necessarily a problem for an original character to be a superhero if she's surrounded by other superheros. Even a character being somewhat extraordinary isn't always a problem after all, we're often more interested in extraordinary characters than dull, normal ones, especially when heroics are required. After all, many canon characters are extraordinary, that's why we like them. But that can also be the first step into turning them into Canon-Sues.