All this talk of Western animation and no mention of WB's Golden Age of Cartoons?
What of 'Pinky and the Brain', 'Freakazoid' and 'Animaniacs'?
This might just be my nostalgia talking, but particularly with the old WB cartoons, I still find them hilarious to this day. They were topical, informative (The Geography Song!), and had jokes for both the kids, and adults. There's also older Disney ones like 'TailSpin', and 'Darkwing Duck', which were fantastic in my books. Again, could be enjoyed by adults. There are a few examples of western animation that is not aimed at children or families. It's French, but the film 'Resistance' is an excellent example.
On the other hand, I've watched my fair share of anime. I think part of the issue many have with it is the cultural barrier. I started off with fairly non-Japanese anime (As in, characters, settings, and culture are decidedly -not- Asian), which I believed helped with the transition. It's gotten to the point now that most cultural aspects don't phase me. These include the points Dan mentioned such as the giant robots, over-the-top fight scenes, and the school girl thing, so it might just be an adaption one has to make.
The reason for the extremes might be due to the fact animation seems to be more so accepted as an all-ages format in Japan, then in North America. Though we have cartoons such as 'The Simpsons', most I talk to assume cartoons equals children or family friendly, particularly in the realm of film. Anime on the other hand can stretch a wide range of topics, including adult themes.
The reason is that anime is not a genre, it's an art style, just like Western animation is. But unlike Western animation, does not limit itself to one audience (children), and is not limited to one topic (comedy, romance, drama, ect), because it's simply a way of animating characters. Admittedly, some genres get milked to the point the udder is dry though (Aka, Giant robots). And perhaps it's a lower bar of entry as the ratio of 'garbage to good' seems to be worse in Japanese animation, then Western.
Final word though, there certainly is good Japanese animation, just that there is also alot of crap animation too. Western animation is very similar in that you can find alot of good animation, just there seems to be less animation in general, so there's not as much trash to sort through to find the good bits.
EDIT: Oh right, gotta recccomend something...
Anime: Last Exile, Planetes, and Baccano!
Western: Everything I mentioned above!