Anime is not better than cartoons. In fact, considering the quality of its animation, it may not even be as good as the older, traditional cartoons. It depends on what shows or movies I'm thinking about, though. Generally animes like to be animated on two to fours (meaning 6-3 frames per second), giving the animators a huge break. But perhaps the same could be said of all Saturday-morning cartoons, which are produced as cheaply as possible.
Still, you can tell that artistically, there is a huge difference between something like NARUTO and Richard Williams' original rendition of The Thief and the Cobbler--not just in artistic style but in effort, too. The Thief and the Cobbler was meant to be Williams' magnum opus, took far too long to produce (31 years), and is still unfinished. Naruto is this anime about ninjas that spans for, what, 500-some episodes and a couple movies? And it's all produced, as of right now, in eight years, nearly four times less than the time it takes for T&C to be made. When you compare the fanbase, it seems to me that Naruto has far more rabid fanboys/fangirls than Richard Williams' masterpiece could ever hope to accumulate. Williams is a genius as an animator. The creators of Naruto and popular animes are geniuses at advertisements and appealing to people.
What is a cartoon? Is it cheaply produced animation with a never-ending, highly appealing story meant to steal the souls of viewers? Or is...is it a-art?