Perhaps I can shed some light on this; I'm not the oldest anime fan but I've been around long enough to see its fandom turn into something entirely different from what it was.
America used to view animation as something for kids only. Kids are stupidly easy to impress. After 20 years I watched Thundercats and the opening sequence still rocked but the dialogue was contrived, the characters were flat, the plots were simple (and contrived to resolve in 20 minutes), and for all the weaponry involved combat was pretty weak--no blades cut skin. But at ages 6-12, we loved this crap. We watched those 30-minute toy commercials and begged mom for everything. Then we moved on to better stuff before you could say "7th grade."
That was America. Japan had no such view on animation. Japan didn't have America's billions and was open to telling stories through animation because it was cheaper. Japan had a culture where the visual novel (manga) was a valid, professional way to relay information and stories. Japan never went through anything like America's 1950's where the Senate brainwashed parents into thinking comics "rot the brain." Japan also has a samurai-rooted culture and the side effect is that half-assed work is not acceptable.
Dragon Ball Z went underground in the states around the mid-90s. It did things no American cartoon had ever done before. People FOUGHT. Not that bullshit where a stray laser beam hits a cave wall and rocks separate heroes and villains. Not that bullshit where someone trips, gets caught, goes to jail, and escapes. Dragon Ball Z was the first time I saw what high-powered combat REALLY might have been like. They punched HARD. They BLED. And you know what else? They CURSED. Animated characters displaying human frustration and rage--things kept off the stage by America's "protect the kids (from shit we do)" culture.
And DBZ was the weaker one. Bubble Gum Crisis 2032 deals with life after the apocalypse, corporate greed run amok, and having to face your friends on the battlefield. Gundam shows war without claiming one side is "evil," but taking the more realistic route and showing that both sides have opposing viewpoints. Ranma 1/2 is one of the funniest things you will ever see but America is to anti-transgender to ever welcome it to television. (Japan doesn't have Jerry Fallwell.)
Seeing that there was animation that grew up with me, I had to get my hands on it. When you have to get something that is hard to get, you need DEDICATION. You have to be clever. You have to learn who has what. I even got a book and learned some Japanese so I could find things more easily. We had to band together. There was no Internet, Facebook, Myspace, or cell phones. It was damn hard to find other people that even knew about anime, let alone had something you didn't. (I airbrushed Goku on the back of a denim jacket. That got noticed only by those who knew it. I can't do that now; everyone knows Goku.)
We formed clubs. Communities. When the internet came up we created info sites. We put conventions together. We told as many people as we could about this great stuff that wasn't like anything you got in America. We got jobs and became consumers. WE CREATED A CUSTOMER BASE THAT MADE IT WORTHWHILE FOR CORPORATIONS TO SPEND BIG MONEY ON LICENSING AND PRODUCTION. That was the end of the fandom I knew.
Circa 2000 the flood gates opened. DBZ and Pokemon hit TV, then Yuugi-Oh. A generation many times larger than ours was introduced to anime not as this wildly different thing that was hard to get, but as something cool on TV. You turn it on and it's there.
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History lesson over. So what's the big deal about anime?
It's the Japanese way of using animation to tell a story. Beyond the style of art it involves rich characters, an intelligent handling of the subject matter, and treating your audience like thinking beings instead of simple children. Avatar is anime. Its creators have adapted Japan's way.
Even Pokemon has deeper levels. Pokemon is about attaining personal growth by working to understand the different beings around you. Ashe doesn't succeed until he learns to trust and understand Pikachuu. The original Japanese opening theme explains this (in metaphor), but was replaced by America's marketing jingle "Gotta catch 'em all."
Edit: This is not to say Japan is incapable of producing cheap bullshit to sell to the kids. Only anime's best and brightest (or at least what is deemed "sellable") get picked up by corporations and sent across the planet. When we say "anime," we're talking about that "best and brightest" subset of all animated material from Japan.