Japanese animation, or anime is becoming increasingly popular in the West. People unfamiliar with the genre might call it cartoons - same as the animation produced in the US or Europe. Indeed, the word "anime" itself means simply animation, or cartoons, in Japanese. So why do fans insist on clear distinction between the Japanese anime and Western cartoons?
There are a number of good reasons why we should call Japanese animation works anime and not cartoons. First of all, anime can be targeted at all age groups. There's anime for kids, sure, but there are also anime movies or series with a lot of mature content meant for older teens or adults to watch. On the other hand, cartoons are a media mainly targeted at children.
Second, anime has a distinctive art style that separates it from cartoons. It involves big, expressive eyes, colorful hair, stylized display of emotions (nosebleeds, sweatdrops and so on) and more. Of course, that isn't to say that all anime look alike. There's lots of different anime art styles, just like there's a lot of distinct cartoon looks. Lately some Western cartoons started copying anime style with some success.
Finally, unlike most cartoons, anime has complex, continual plots. A cartoon is often self-contained in a single episode, e.g. you can watch the episodes in any order and still enjoy them. On the other hand, you have to watch anime series in order to understand what's going on. Some of them span for hundreds and hundreds of episodes.
I am just trying to say that anime and cartoons are vastly different, so using a different term for each is completely justified.