The core problem with this whole argument goes back to the original comics. Because comics are a visual medium, long-time comic fans have an established idea of what the different main characters look like, the designs of their costumes, their origin stories and personal history. Although many of these characters have been re-booted and gone through various costume changes and alternate time-line/universe story-lines that played around with changing some of those core concepts, when a comic fan thinks of a specific super hero, he usually has a pretty specific mental image in mind on what that character "should" look like and how he/she should act in order to be as "true" as possible to his origins.
When porting these characters over to a new medium, like cartoons, books, TV, movies, audio-books, or whatever, a fan of the character is going to be looking for a representation of the character that comes as close as possible to the source material while adapting, as needed, to the new media. Sometimes the character is ported over as a very close match and other times the adaption changes quite a bit, without much justification beyond artistic license. Hardcore fans tend to be VERY resistant to any change to an established character because they already appreciate the character in his/her current form. Any change might threaten that appreciation or make the character so different that it might as well be a completely new hero rather than a re-boot of the original. To give a well-known example, Batman has gone through various incarnations over the years, from the campy gadget-using Silver Age Batman of comics-past to the much more sinister and gritty Dark Knight of modern movie fame. If you compare these two versions of Batman side-by-side, there's not much similarity. They could easily be two unrelated heroes, but they are both "Batman".
A lot of the people who complain about changing the race or gender or hair color of a superhero in a movie/tv adaption are not being intentionally racist or sexist or ... um .. hair-colorist? They are protesting what they consider to be an unnecessary alteration of an existing character which might lead to an unfavorable change in the overall image of the character from their personal perspective. If you change one of the character's core features, then you might feel it is no big deal to change another and another ... and another. Eventually, enough things change so that superhero stops being the same character, other than sharing a name. Even if the new version is pretty good and well-received by the movie's general audience, longtime fans will notice the difference and it will bother them - in the same way that people who have read the book a movie is based on often come away from the movie feeling cheated because certain scenes were cut or characters were altered to better fit the screenplay. If you are unfamiliar with the source material or not emotionally-invested in it, then it doesn't matter if Peter Parker works for Daily Bugle as a freelance photographer or if he works for a teen fashion magazine as a freelance writer. Yet to a hardcore fan, it would completely change the character, since Peter taking pictures of himself as Spiderman for extra income was a long-running side story in the comics and some of the movies.
Does it really matter? Probably not.
Does it REALLY matter to a long-time Spiderman fan? Hell yes.
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