The music: Alternative and indie are similar, and both came from college rock, which is what happened when goth became more mainstream (and goth was what happened when punk became more mainstream). The difference between alternative and indie is simple. Indie bands are on an independent label (or unsigned) and alternative bands aren't. Thus, alternative is more 'mainstream' than indie.
The...other thing: Like most music, people who listen to it are sometimes labeled by it. Alternative and indie are once again similar, but often times indie kids are contemptuous of alternative kids because they're mainstream, which is stupid because the whole point is the music, and is also the reason that punk and goth etc. died because they became 'mainstream'. Thus, sometimes indie kids are obsessed with not becoming mainstream, and consider bands that sign with a major label and the people who listen to the sellouts. Alternative kids usually care less about that (even if they listen to indie bands) and listen to the music that they like, even if it isn't indie or even alternative or rock, the music they like is the music they like. Like indie kids though, alternative kids will never label themselves that way, and no one will ever label them that way either, except to say 'people who listen to alt'. If you label yourself that way, you're probably just like someone who labels themselves punk or goth, trying to be 'unique' instead of yourself.
Alternative kid: Sure, that sucks and all, but I still like their music, so I don't care that much.