Even the worst anime is better than most American cartoons, and I often find myself asking why. I think I've found some of the answers, though I'm sure there are many more reasons.
First off, anime is better for older audiences. Americans have some wierd mental block where cartoons are only for children. Even cartoons that have humor that can appeal to adults has to "wrap" that humor up in slapstick, plotless cartoons that a child with the attention span of gnat could enjoy. There are a few notable exceptions, such as the Simpsons and Daria, that appeal to the teen/adult crowd, but they are very rare.
American cartoons also have very little plot. All American cartoons are episodic, with either very little plot, or no plot, carrying through the entire series. Even the adult cartoons follow this American rule. Many Japanese cartoons are also episodic, but even those have a continuing plot, and missing one crucial episode can leave you confused the next time you turn on the show. A large number of anime aren't episodic at all, and they often leave you with suspenseful cliffhangers. This isn't a big problem for foreign fans because they often buy episodes in groups, but I can tell you from experience, now that I live in Japan, that waiting for next week's episode is just as horrible as any cliffhanger in an American drama. Anime plots are usually absorbing and often epic in scope.
Which leads right up to another reason that I love anime. Many anime plots are fantasy based, and an equally large number have sci-fi plots. Cartoons are the perfect vehicle for fantasy because they give a good visual representation without the expense you incur trying to do live action fantasy. Even Willow, the best fantasy live action movie to date, was pretty cheesy at times. (Okay, I wrote that before Lord of the Rings or Harry Potter came out. Those are both good examples of live action fantasy...but they're still far and few between.) It's difficult to do fantasy well in live action, and often prohibitivly expensive. In America, good fantasy cartoons are few and far between. You get the occasional Disney movie, and that's about it. But in Japan, you can turn on the TV in the early evening and you have a good chance of having a good fantasy adventure. While there are many more science fiction movies/series available in the West, there are very few cartoons, and they all have one problem...little plot carryover. Anime sci-fi, on the other had, can get very in-depth because of the length of the actual plot, though each episode may be rather short.
My recent anime favorites tend to have a lot of Japanese history in them. I liked Mononoke Hime, I loved Rurouni Kenshin, and I adore Inuyasha. Now these are either legend/myth type stories or "fictionalized" history, so the accuracy is not a hundred percent, but it's still reasonably accurate, and the customs and traditions that the Japanese accept as a matter of course are new to me. I love looking at the pictures, costumes, and listening to the speech of the characters in Rurouni Kenshin becuase it's so different from my experience. The legends and characters in Mononoke Hime and Inuyasha are interesting and fresh to me, and it greatly deepens my pleasure in watching the cartoons.
I have been harping on plot and format, but another reason I love Japanese anime is because the quality of animation is usually very high. Every series (except Inuyasha and Escaflowne) has a bad episode or two, where some studio lost track of good character design or experimented too much with strange perspectives, but overall the animation is heads above American animation. The scenery is detailed, the characters are smoothly animated, and you can't see when the background is going to move because the color is horribly, glaringly different.
The voice acting is also usually better. One reason so many otaku hate dubs is because even American/Western professionals don't measure up well against most Japanese voice actors. The depth of emotions, and their ability to convey different emotions and jump from drama to slapstick comedy in a matter of seconds never ceases to amaze me.
The music is also unbelievable. Usually the opening and closing themes are famous pop music hits, and the background scores are specially made for the series, and wonderful. Each successful anime also has many extra songs that never played in the actual series recorded on special collector's CDs. Some especially amazing series, as far as the music is concerned, are Escaflowne and Rurouni Kenshin. I enjoyed Escaflowne, but the music MADE that series, as far as I was concerned. I have all four CDs and I LOVE them.
Well, I hope I've convinced you to give anime a try. Or I've confirmed some of your own suspicions of what makes anime so much better than western cartoons. If you have never watched anime, give it a try. I have yet to watch a series that I didn't enjoy!