To start off, I lived in Indonesia for a great deal of my childhood, where like most East/South East Asian countries - if not all of them - have youths who grew up watching anime, most likely dubbed. So I got hooked into that.
But why particularly anime? For me, the great appeal was the animation which i found more appealing than most other cartoon, a lot of it was more dynamic, the action sequences were better choreographed (for lack of a better word) than most other cartoons. Yeah, I was into that stuff, i got my anime fix when i was a teenager from AXN, like Samurai X and Flame of Recca. Certainly, I enjoyed other cartoons, like Batman: the animated series, but the other cartoons that was of the action genre would look more blocky and even static.
I feel that you can tell more of a difference between Japanese and American cartoons now than, say 20 years ago. Well, except that even back then, animes were still pushing the limits on violence and sexuality than American ones. I'm just thinking about Saint Seiya at the moment, and comparing it to the Thundercats. Man, they were both great shows from what little I remember of them.