So, why should you watch anime? That's a tough question- even tougher to explain to those who are uninterested in animation. I'll try to explain it from both angles- as someone who watches cartoons and also as someone simply looking to foray directly into the genre.
As a subdivision of animation, anime covers a lot of ground. Simply saying that it is in a league of its own would be wrong- there are many American shows that rival anime in story, scope, and art direction, as well as theme and structure. What you need to understand, first and foremost, is that anime is merely a label for Japanese animation. They have eleven minute episodes. They have long, spanning shows. They have miniseries, something we don't usually see in American animation. They just have a far more distinct visual style than we do.
Staples of anime are pretty much what everyone expects- odd hair color, big, bright eyes, bulging muscles, and fantastic monsters both cute and monstrous. The art style sort of sets these stylistic choices up, really. The over-exaggerated proportions lend themselves to a lot of what you see in the genre. However, "what you see in the genre" is a very broad term.
It is important to remember that anime is a deeper and more appreciated form of art than how most western animation is viewed. Some anime, like Dragonball, Gundam, Bleach, and Fullmetal Alchemist have deep-seated appreciation in both western and eastern society. Anime like Neon Genesis Evangelion is such a sensation that the retelling of the series as full-length theatrical releases is considered a huge event. A full-scale Gundam statue was erected in Odaiba's Shiokaze Park. Gigantor, one of the first giant robot anime, has received the same treatment. These pieces of animation are beloved because of the stories they tell- which is often a recurring theme in most popular anime.
It seems that the ability to tell a concise, fantastic story is a common thing in anime- and most continuous series attempt to end their story and begin a new one. Neon Genesis Evangelion, Fullmetal Alchemist, Trigun, Gurren Lagann... they have all had a limited run in which they tell a specific story. Gundam is one of the few franchises that, after finishing its lengthy story, moved on to tell a new one- creating a whole new timeline, characters, events, and the like. There are a few series that pride themselves in telling a continuous, spanning story- mostly because they appeal to a younger audience. Dragonball (Z and GT), Bleach, and Naruto are especially lengthy series, with the latter two showing no signs of stopping anytime soon. This is perhaps because this form of animation is so well-appreciated that the projects are given so much free-reign in design and writing. It's hard to really say how anime got to such a point, but it does stand as a testament to its ability when it has decade-spanning franchises that are so beloved.
And not based on selling toys.
In any case, you'll find a great deal of shows within anime- silly, random, slapstick and dialogue driven comedies, parodies on well-established franchises, new takes on classic stories... pretty much everything. There are big name anime projects and smaller, lesser-known projects. However, if there is one thing that I particularly enjoy about anime is the more-lenient censorship- I really don't mean to put a negative twist on that, but that's really what it is- atop more mature storytelling.
If you're a fan of cartoons but desire something a little meatier, anime is like graduating from elementary school and moving in with the bigger kids. with more mature themes and adult material (I mean that in the most innocent sounding way possible- I'm not talking hentai, here), anime is chok-full of engrossing characters, stories, bizarre twists, and gratuitous violence. That's simply putting it as plainly as I can. Action serves a purpose in anime, and it is not to be tossed away by stock flashes and to look cool- it moves along the plot. That's why, even in anime that you HEAR about having a lot of action sequences, they usually don't- they're usually just very memorable.
I can't really say anything else about anime- although another trait to be aware of is the verbosity of the characters- they're often extremely profound or extremely stupid. However, my recommendation is that you watch a sub of an anime- where the Japanese voice acting remains intact with English subtitles. Japanese really is a fantastic, emotive language.
I suppose recommending my favorite cartoons and anime at the same time was a little bit of a hasty step- seeing as the two are so drastically different on so many levels. I will return to this subject and specify what cartoons I like and what anime I like- they're worlds apart. So, I apologize in hindsight. But anime really is an art form- which is why, I can only assume, it is respected by Japanese culture so much. Not saying that American cartoons are NOT (I'll be getting to that in another section), but anime is at least identified as such... very often.