It's a pretty dodgy proposition to define who is and isn't a super hero, but let's give it a whirl. First off there's the semi-obvious stuff: greater than normal abilities. The vast majority of super-heroes have some ability or combination of abilities that are paranormal in some respect. Superman, of course, is the prime example. Even your Batman, Nightwing, Hawkeye, Mr. Terrific, etc. type characters are, generally, so far above anything normal people could do as to be effectively paranormal. (Batman--gadgets and fighting skills aside--is the world's greatest detective. Hawkeye is the world's greatest archer. Nightwing may not be the world's greatest acrobat (it's up in the air, I'd say) but he's so far above anything you'd see at Ringling Brothers as to make the argument moot.
Secondly, the costume and colorful moniker certainly helps, but isn't particularly neccesary (ref: Luke Cage)
But really, it's the attitude that makes the difference, the motivation. There's a sense of altruism that's caught up in that super-herodom, a--as Spider-Man put it--a sense that "With Great Power Comes Great Responsibility." Batman's a loon, on one level, but Superman had his number cold in JLA, when he pointed out to him that, underneath it all, Bats was someone who wanted to make sure that no one ever died the way his parents did again. (And, see, this is why I love JLA when it's working--because only Superman would have the balls to call Batman on being a softy at heart.)
So, really, I'd say it's a combination of these factors. Of course, with any good literature, there are exceptions and gray areas. Let's look at them:
- The X-Men: Are the X-Men super-heroes? Certainly, they have a lot going on, in that department, although that wasn't really Prof. X's goal at the time. They were trained for a very specific mission: stop Magneto from conquering humanity, protect humans from "evil" mutants, and serve as a sort of PR stunt for mutants. As Grant Morrison wrote, being a super-hero was something people could understand. This also seems to be something Joss understands, as he's quite consciously pushing them back in that direction. That being said, I think the X-Men can quite honestly be said to be soldiers who sort-of became Super-Heroes along the way. Plus, on topic.
- The Punisher: The Punisher was intended, originally, to be more of a dark reflection of the super-hero than to actually be one himself. He's a guy with nothing but a bunch of guns and a costume, who's pushed to kill criminals out of vengeance. It's similar to, say, Batman, but the fact remains, The Punisher's been fairly cavalier about who he kills--at points offing jaywalkers and litterers. So, no, he doesn't get a pass from me. Interesting character, certainly. But not a super-hero.
- Buffy the Vampire Slayer: Hmm. Super powers? Check. Altruistically putting others before her own safety? Check. (OK--she's had her selfish moments here and there, but that's fine. Nobody, not even Superman, has to be perfect, and I think she's hit more than miss in the selflessness.) Costume? Not so much, but I think she can get a pass on it. Buffy's a super-hero. Think the same holds true for Angel, although most of the rest fall into the "faithful allies" category.
- John Constantine: John has some magical abilities, but most of it is just bluffing and image. No costume, certainly, and really, his regard for others' safety is dubious, at best. Man's got a body count like 'Nam around him. He does a lot of good, but really? I think he's be much happier having a pint over at the antihero bar.