Both anime and cartoons fall under animation as its a blanket term.
Cartoons came first and anime artists in japan copied the americans animation styles(thats why if you watch anime you’ll notice that most of the characters actually look american rather than japanese) Generally when i think of cartoons i think more of a kids show, but thats clearly not always the case.IMO most cartoons on tv now are mindless where as the cartoons of the 90’s were great.(rockos modern life, freakazoid,ahh real monsters,earthworm jim,ect)
On the other side of the world we have anime which generally appeals to the same audiences as mangas(japanes graphic novels/comics) as most animes start off as a manga. Much like here in america where comics will be made into cartoons.
Generally anime is japanese but that doesnt mean america doesnt try their hand at it.(usually failing miserably) There is probably an anime out there for everyone (beings you like cartoons that is)
Some anime as great, some is horrible. Personally i hate when anime gets dubbed, id rather just watch the japanese version with english subs because the voice actors are usually god awful. For instance Bleach and Death Note. Both great animes but when Adult Swim got their hands on them they completely murdered them with the voice acting. Thats not to say all english dubs are bad though. Hellsing, Cowboy Bebop, Trigun, and FLCL all have good voice actors.
Anime also tends to show over exaggerate facial expressions alot.