I haven't seen Neon Genesis Evangelion, but I can answer that one.

Unlike american-liked series, which tend to go on and on until the public's interest wanes, Japanese TV series tend to have a clear beginning and end set from the start of the project. As such, the creators are free to build a story as solid as possible, since they know exactly how much space they have to flesh it out. You'll find that several great anime shows follow this formula.

Then explain Dragon Ball, Sailor Moon, InuYasha, Naruto, or other shows with 100+ or even 200+ episodes? I really doubt these shows need so much filler for a storyline.

I applaud series that are under 50, 20, or even under 15 episodes. Instead of rambling on like most animes do, they have a clear purpose and story. I never really liked Neon Genesis Evangelion, but I can easily tell someone what the themes and meaning of the show are. Many shows just continue with a story just for the sake of continuing. They might switch the goal of the characters a few times, delve deeper into some sidecharacter's life, or go off a tangent like "Let's go to the beach for 3 episodes!" or "Although you haven't seen me go to school before, you will now for 10 episodes!" to hide the fact that they are tired of the original premise but also too lazy to make a new series with new characters. For example, instead of saying, "some magical girl collects Lollipop Photos from monsters that corresponds to all the colors of the rainbow for 10 episodes while the rest of 90 episodes are her trying to find a boyfriend", you can say "this short anime is about life and death, the consequences of freedom, and a classic tale of growing up."

I'm not saying under 20 episodes necessarily makes a good anime. It just tends to be better written and manages their episode budget better. But to make a great anime series, that requires a great director or writer with a budget that can see through the entire project. The low amount of episodes just complement that.

When it comes down to it, anime isn't as far away from American cartoons. They just have more fan service to compensate.

I said "tend to", and not "all follow the same pattern". Of course, there are exceptions on both sides, and not all studios want to stick to some formula. Still, you can find that this applies to several series.

Even the Rockman.EXE anime, which lasted for 209 episodes, introduced a new major concept every season or two. As such, each main story was resolved in a set amount of time, making things fresh once again later on.