For Marvel purposes, a mutant is a being who possesses a genetic structure not present in their parents. Obviously useless as a scientific definition (otherwise, all non-mutant children in Marvel would exactly resemble their parents, like clones), it’s mainly used as a tag for a particular unit of superhumans.
Actually, the definition is a bit looser than that, since following any form of realistic biology is usually not the top priority for the writers. For instance, Siryn, the daughter of Banshee, is called a mutant, despite the fact that her powers are the same as her father’s. Attempts have been made to say that Siryn has a “different” mutation, in that she can talk and scream at the same time (don’t worry, it makes sense if you know the characters), but the main fault, obviously, is bad writing, not bad genes. The easier way to define a mutant is that they have an active X-Factor, and that’s really the main point of the whole definition. You will see references to “a mutant has to have a different power than their parents” in mutant comics, though, so it’s mentioned here just to get you acquainted with it.
So, what are mutants, then? They are superhuman because they were born that way. They didn’t need any gamma bomb blowing up, or spider biting them, or magical formula recited. They’re superhuman because that’s the way they were born to be. They are mutants because of their X-Factor. And what is an X-Factor? Read below, true believer!
The reason there are mutants on Earth comes from Marvel cosmology. Large, alien gods, called Celestials (who some say are but the incarnations of the dreams of Eternity) visit all planets that will bear life, early in the planet’s existance. There they perform genetic tinkering with the early life that will, if everything works out right, leave the species on the planets with three distinct superhuman bloodlines: Eternals (who never suffer random mutations), Deviants (who always suffer mutations in each generation), and the normal folks. It’s the normal folks that concern us, now, for in them the Celestials left a special genetic trigger that would allow some normals to gain superhuman powers after exposure to odd events (like the Fantastic Four, the Hulk, or Spider-Man), and in others would self-trigger when exposed natally to sufficient background radiation which would make the child a mutant. When it’s self-triggered, that genetic trigger is called the X-Factor.
Now the X-Factor only makes a mutant when it’s self-triggered. Something happens to it when it does so that it becomes different than the same gene that allowed Spider-Man or the Fantastic Four to gain their powers; mutants show up on mutant detectors (which look for the unique signature of the X-Factor), while Spider-Man doesn’t. Mutants also give off unique brain patterns due to the X-Factor that enable telepaths who know what to look for (like Professor X) to detect mutants far more easily than normal humans or non-mutant superheroes. Devices that nullify mutant powers by negating the X-Factor are useless against non-mutants as well. On the other hand, Ship (an old base of Apocalypse) had a force field around it that would only open if it detected the X-Factor inside a visitor. In short, yes, mutants are different than the “normal” superhumans in a Marvel comic. Aside from that, there’s no real appreciable difference or superiority for mutant superpowers over non-mutant ones – very few mutants are as strong as the Thing, for instance.