Anime (at least, Japanese Anime) usually have deeper plots and premises than that of American Cartoons which are primarily aimed at young children. Indeed, there are a multitude of genres of Anime in Japan, not just for young children (sometimes quite the opposite...). So that's how Anime is different than other cartoons.

This is not to say anime is better than western cartoons, as there are a good number of really horrible ones.

Still...I'll have my Death Note and FLCL. <3