Cartoon is a term that once referred to fine art during the renaissance period. Somehow, though, the term's meaning evolved (or devolved, depending on how you look at it) into humorous and witty illustrations and, finally, into animated programs meant to entertain people, mostly children. For, clarity, the term 'cartoon' in this article shall be referring to the animated programs. Recently, however, a new term has been coined: Anime. And the distinction between these two has caused much confusion.
Strictly speaking though, there should be no distinction between anime and cartoon. Anime is a term used by the Japanese to refer to animated cartoons, from their own country and from others as well. Somehow, though, the world started using the term anime in reference to Japanese cartoons exclusively. Japanese animation was once called japanimation, but this term was aptly discarded when the term 'Anime' was put into use.
Some say that they see absolutely no difference between the two, even going so far as to classify both cartoons and anime as 'childish' because of the nature of how they are presented, much to the woe of anime fans. Technically, there's no official difference and they are somewhat correct. But you only have to watch a single episode of any anime series and you'll see that anime is far more than your usual slapstick comedy man-tripping-on-a-banana-peal early morning cartoons. There is a real and profound difference between the two besides their spellings: Anime is so much more than 'just a cartoon'.
Storylines/Plots
Cartoons usually have a simple and defined plot. And they're usually about the good guys fighting the bad guys. Cartoons rarely go beyond that old cliché. Most Anime series, on the other hand, are also about the same good vs. evil but the plots are rarely simple and rarely predictable. You'll find an element of politics, religion, humanity, and a score of several other abstract concepts within an anime. Whereas cartoons are just about 'evil' cats chasing 'innocent' mice or a hero defending himself and others from a very evil bully. Anime can blur the lines between good and evil so much that you'll be left wondering which side is really right. Look at the plot of Gundam, there aren't any good guys or bad guys.
They're both fighting for a cause - a worthy one, and both sides are prepared to do good and evil to achieve their goals. Look at the Wave Country arc of Naruto. Zabuza, evil as he was, had evoked more sympathetic tears than any of the typical Disney antagonists. There are, however, few exceptions to the 'cartoons are simple' rule. There are cartoon series that are far advanced like X-men.
In terms of humor, anime goes beyond the slapstick comedy that cartoons offer. There won't be any corny joke about the chicken crossing the road with anime. There won't even be the quintessential banana peal. Anime humor is deep and witty. However, it is to be noted that cartoons are mostly for kids. Anime can be for an adult audience; it mostly is, actually. There are even times when anime plots are hardly suited for children.
Characters
In cartoons, there's a protagonist, an antagonist, and a damsel in distress. Their roles are well-defined and clear and the characters always act according to their roles. In anime, however, the damsel in distress may very well become the antagonist, leaving the protagonist in distress. There's no knowing what to expect in anime. Anime characters, a lot of times, are developed overtime through trials and growth, while cartoon characters remain the same from when the show started. (Of course, there are exceptions such as Spiderman.) Since there's an ongoing plot, Anime characters actually grow up and may even produce the next generation for the series. Anime fans tend to get into the characters a lot deeper because of the characters' depth and charm. Consider Naruto; viewers see him growing up from an infant to a young powerful ninja, the way he relates to people, the way he learns new techniques, and the way he evolves visually. That is one of the chief reasons for popularities of anime over plain cartoons.