A lot of this has probably been said, but I feel like jumping in anyway.
A lot of the American bias against anime comes from our perception that cartoons are for kids. And nearly all American animation is. What Dreamworks, Warner Brothers, and especially Disney, while appreciated by people of all ages, is aimed primarily at the children's market.
Anime, however, comes in a variety of forms. Some of it is aimed at children, while others are aimed at teens and adults. Hentai, a subcategory of anime, is pornographic and its availability in the US, like all other sexually explicit content, is limited to those over 18. Just goes to show that the American perception of "animation is for kids" is not shared in other parts of the world.
All anime that I know of being shown on US television is dubbed. By and large, dubbed anime is terrible. Sometimes the plot gets changed, more often the music. The worst however, is simply the quality of the voice acting. The emotional undertones of the original seems to get lost in the dub.
As an example, I have seen in its entirety, the subtitled version of The Vision of Escaflowne as well as a few episodes of the English dubbed version that aired last year on the Fox network. The plot seemed to be right, but the voice acting was bad, and the opening video and music were completely changed. Also, I noticed that while some of the music had been retained, other parts had been changed. I stopped watching because it wasn't worth my time.
Most American TV animation tends to be very episodic in format. Rarely does a story span multiple episodes, and often it seems that one episode will actually contain several short independent stories. Keep in mind here that I haven't watched too much of the recent American tv animation; I'm basing this on shows like Garfield & Friends, Muppet Babies, Tiny Toon Adventures as well as the classic shows already mentioned like Looney Tunes.
Anime, on the other hand, tends to develop a single overall story arc that gets carried through from beginning to end. This is done to various degrees. Shows like Tenchi Muyo seem to be more episodic, while The Vision of Escaflowne is quite clearly built around a single storyline.
Speaking of storyline, this is one area where anime clearly has an edge over American animation. Being geared towards children, American animation typically has simple plots that can easily be grasped, like the Road Runner endlessly outwitting Wylie E. Coyote. Anime often has much more complicated themes, such as the struggle between animal nature and human technology in Nausicaä of the Valley of the Winds and X. Anime has some thought provoking themes, some things that would provoke a strong reaction from some conservative, xenophobic groups if it were more mainstream.
A few of my anime favorites. If you want to check them out for yourself, be sure to get a subtitled version. Subtitled videotapes seem to be harder to find than dubs, so if you have access to a DVD player, that generally works best. DVD typically offers both Japanese and English audio tracks (and sometimes other languages as well), and you can turn subtitles on or off.
Ghost in the Shell
Nausicaä of the Valley of the Winds
Neon Genesis Evangelion
Serial Experiments Lain
The Vision of Escaflowne
David / SAN