I read to page 5 and didn't see anyone reply really what I think is "up" with the deal of people saying "anime is more mature than cartoons".
The fact is, none is, by default, anything. Animes, just like books, comic books, cartoons, movies, series and even plays, are just a way to tell a story. Not to be an Yathzee fanboy but I recall him, himself, mentioning something in those lines - that anime, is not, for the majority of it, a GENRE. It's just a type of media.
Being so, it's absolutly ridiculous to have people come around and say "Hey anime is about robots and boobs and fanservice and epic, slow motion ninja fights". It's not. A genre of anime might be about all of the above, but at the same time, a genre of movies include crappy made aliens and pseudo-american Steroid-pumped Space marines, does that means ALL movies have that? No. Just no.
In fact, some of my favorite anime have neither of those. Death Note has absolutly no fights, nor robots, nor much magic (Despite the death note, but that's what carry the moral questions of the story).
Rurouni Kenshin OVA are probably the best Samurai anime I've ever seem, yet it contains absolutly no over-the-top fights, no name-calling of moves, no nothing. It's brutal, dramatic and simply amazing.
So while it's 100% wrong to say "animes are more mature", it's also silly to say "animes are less mature". They're just another way to get your story out there.
However, and it's something I'd like to point out, animes are a much more respected way to publish your story in Japan than it is in the West. We, on the west, apparently don't take drawed work seriously just because it was drawed and dismiss it as a "second-rate" media, reserving the great storylines for movies, series, books and plays (and maybe games!).
In Japan, such drawed works have way more cultural relevance and being so, they often are the choosen media for really profound work (Like someone mentioned, Grave of the Fireflies). If the grave of the Fireflies autor was in the west, he'd probably never think about telling his story in a "cartoon", because of social stigma attached to "lol cartoons".
So ends up that yes, there are more animes with a serious plot and story, more than western cartoons, but that's not because this is a "serious-only" kind of media, and instead, just because people on the East accept it for what it is - just another way to express yourself, free of any attachments to what an "Anime should be".