The main difference between an Anime and a Cartoon is that by definition: Anime is a particular style of animation that originated in Japan. One could even say that ALL computer animated shows that is/was made in Japan are to be called Anime. Cartoons on the other hand are what the Westerners (US and some EU contries) call animated films and shows.
In short: If the animation originated from Japan then it is called Anime, otherwise called Cartoon (especially if made on the Western world).
Another notable difference between the two is their target audience. Cartoons are mostly targeting babies-to-young children while Anime has a large target demographic that even includes adults.
You will often see no violence or romantic side-stories from cartoons. Tom and Jerry, Dora the Explorer, Looney Tunes and Barney are all cartoons that only talk about friendship and having fun. Which is also the reason why Cartoons often use "talking animals" and simple colorful graphics to make it easy for young children to differentiate what's what. Basically it's for babies lol Although Cartoons have been slowly upping their age target demographic to include young adults like the Cartoon Avatar.
While Anime are for all ages and is not limited to any genre. Anime could include love, death, betrayal, conflicts, action, violence and more. Basically one could say that Anime can go deeper in terms of character and plot development unlike Cartoon's simplistic approach to almost anything.