@Perpetual Lurker
Anime and Cartoons are the same idea in concept... but so are a car and a truck, or for that matter an airplane and a train. They accomplish similar goes, but the devil is in how they go about it, what thinking and mindsets go into it.
I think a large part of the problem is that the internet has perpetuated a very shallow view of what "anime" is. It's only here, for example, that pretenders like Avatar and Teen Titans can pass for the real thing. What highlights the difference is, basically, the attitudes behind the stories and the approach to the format. Western (or specifically American) animation still hasn't really lost the radio-show inspired "explain what we're seeing" style of writing. Granted, its gotten less obnoxious, but you still have to have Aang highlight what his plan is before he executes it so the viewer is sure to follow the action.
That, to me, is the big difference:
Western Animation is obsessed with explanations, and making absolutely sure that you understand what's happening and there is no ambiguity. Anime on the other hand often relies on you to just "get" things, and doesn't care if it leaves you a little lost. I said earlier that I recently discovered
Gensoumaden Saiyuki and the second DVD volume (episodes 6-10) are perfect examples—you often have no effing clue what's going on with the heroes and whether they like each other or not (a viewer with low intuition might even think they hate each other's guts), and the show never comes right out and says "these are a bunch of good friends acting tough because they're comfortable showing their worse sides to each other." It just lets you see that for yourself.
I think what often confuses these discussions is that they always become about "Anime Good, WA Bad" and people become defensive over that, and it cancels out all thought. I hope that doesn't happen here. The WA style of overly explaining things has its place and there are a lot of stories where it really works. It just needs to be pointed out that
this is the monumental difference, and why shows like Avatar aren't quite anime even though they try their best to look like it.