I've grown up watching both and I like em both equally.
I do enjoy Japanese anime more when it comes to storylines. Most American cartoons are made specifically for dumb children so it is hard to find one that has a good story. Most anime are more mature and also have different types of stories from American shows. They range from action, romance, sci-fi, mecha, fantasy, slice of life, school life, drama, cooking, music, action, adventure, sports, martial arts, and psychological, etc,....
When it comes to the animation, I like American better. Japanese anime is very well drawn, but not well animated. The characters are often too still, while american animation will have the characters moving and have some body language when they talk. But I do appreciate that anime keeps the lips only up and down, keeps the lip syncing from looking awkward when they get dubbed. But nothing can compare to the animation quality of the classic disney films, ahh so beautiful ^_^
But really it's hard to compare such different things. Even if something is less realistic or whatever it can still be a good and fun thing to watch.