Anime refers specifically to Japanese animation
Anime is a term coined in Japan, shortened from the word 'animation' and specifically used for their animation style. While anime is sometimes referred to as a cartoon its main difference is its origins.
Anime is short for animation in Japan and started to be used in the 1970's, though animation had started about 70 years prior. Animation, while encompassing many forms or art and media people see today, anime is a bit different. Partially from the art style used, and partially from the plots they have. Anime has a higher focus on the emotional side of things, be they relationships or problems. The history of anime is rich and over time has been cultivated into an entire culture with Anime Conventions, even coining the term 'otaku' for people who are obsessed with it. The history of cartoons though, is more historical. Cartoons were illustrations before they graduated to television. Political comics before kids shows. Those who love comics and the like are considered geeks, the term less offensive than it was the previous generations. Both have their own distinctive fans, just like they have their own distinctive book versions and screen versions. Comics and manga, cartoons and anime. They're different categories, with different fans, and different origins.
The term anime, while created directly in relation to Japan's animation industry being on the rise in the 1970's. In truth, there's no special reason behind this except the fact it's a shorter, faster way to say animation. Anime, or アニメ(A–Ni–Me), would be アニメーション (A–Ni–Me–Sho–N), longer, and rather unnecessary when the first is just as understandable and recognizable. It was a term created for its ease, quite like how in English, people will say ASAP instead of 'as soon as possible.' You can use the full version but usually there's no need to do so. When traveling to different regions there's usually a style that's favored. Be it with colors, food, or art. Art in particular stands out. Italian art contains a lot of frescos, paintings done on wet plaster, while European art contains a lot of oil paintings. With time, artwork and their techniques evolved, till animation became another form or art. There are different types of art in the world, belonging to different places, different people, but that does not change the fact that they are art. Much like with animation, they are made in different places by different people and called different things, but they are still considered animation.
[P1] There's different culture behind cartoons and anime, shown through their fans