Maaaan, this is a big discussion with a lot behind it. On that note, please forgive me if this is a big post (and if I'm bumping this thread, just kinda had to throw my hat in):
So, I have a long, storied history with anime, full of ups and downs. When I was a kid, I grew up on things like Dragonball Z and Digemon. There was a point though where my parents took issue with the 'extreme' violence of Dragonball and wouldn't let me watch it. I was also very easily swayed by peer pressure, with the prevalence of kids my age mocking the show for its eccentricities and my bitter resentment for not being able to watch it combined to make me hate the show and by extension anime in general (I also used to love Samurai Jack until some kid said it was for "babies", which is so laughably ironic in retrospect I can't even put it into words). That would change a couple of years later attending summer camp in the limbo between grade school and junior high, when it was announced that we were going to watch Spirited Away. I scoffed at first, but as you might imagine, I kind of had my little 10 year-old mind rocked. Not only was it really good, but it was surprisingly different from any of the stuff I had seen before. From there I discovered Adult Swim a few years later, and was enamored by stuff like Fullmetal Alchemist and Eureka Seven (stuff that I still appreciate today).
And Adult Swim is STILL a big source of my anime viewing, as I never really got that much into the Eastern culture as a whole--despite growing up with Nintendo as well. My obnoxious, weaboo "friends" in High School didn't help things much, though I still do like eastern stuff today. My anime-appreciating college friends (NOT weaboos) are considerably less obnoxious and occasionally turn me onto some good shows. The one thing that's struck me about people who can appreciate two entirely different things is that they often are able to translate everything they see through a personal lens. It allows them to empathize with and understand that 'other' from the point of view of what they're all about. A good example of this is a friend of mine in High School who was very much immersed in the Hip-Hop culture and enjoyed mainstream American movies and games, but also loved Disney Musicals. And let's be clear, he was the farthest thing from the traditional nerd archetype that we're familiar with. Speaking of which, if you gentlemen would allow me to talk about 90's 'gangsta-rap' culture for a moment, I can continue making my broader point, which is: influences are weird and often go both ways. Many historically significant rappers today, such as Snoop Dogg and members of the Wu-Tang Clan grew up on Eastern Culture. A lot of their music then took inspiration from Chinese Kung-Fu flicks and Japanese classics, specifically Seven Samurai. Hell, the Wu-Tang Clan named themselves after the 1983 film, Shaolin and Wu Tang, directed by the Chinese film actor and martial artist Gordon Liu. Quentin Tarantino was a big fan of his work, as shown by Liu's role in Tarentino's Kill Bill movies. I think it's safe to say that eastern cultures influence the west as much as we influence them. That said, anime often doesn't focus on Japanese culture, which is a little disappointing considering how rich it is. Games like Muramasa: The Demon Blade and Okami gave me a great appreciation for such things.
I also think it's fair to say that anime is a markedly different beast to Japanese films particularly, as well as other forms of media. In fact, I've come to the personal conclusion that anime as a whole operates within a pulp genre, like, say, comic books and science fiction. Exceptional works of art may come out of these mediums, but they are almost undeniably that, exceptions. And even those few shining examples come with the trappings of their genre (which may not be an inherently bad thing). Though I'm able to admit that I'm not all that well versed in the rest of the culture with the exception of games (though I have watched quite a few Korean films, Japanese movies and live-action television has eluded me). One big personal exception for myself (in terms of anime anyway) would be One Piece. I love this damn show, for various reasons. The storytelling, though it takes place in an absurd world, is very much grounded in reality. Through a colorful and incredibly creative and diverse cast of characters, the show explores themes of psychological trauma and depression, racism, political corruption and so on. I find it incredibly relatable, and maybe that's because it's a series with western sensibilities, or perhaps it just taps into something universal.
PS: That and it definitely operates on a rule of cool, which seems to be something a bit more popular in the western world. The admirals and their designs, Sanji and Zorro working together for just. ten. seconds, and the whole scene from Ennies Lobby where the Straw Hats proceed to walk Reservoir Dogs style up to their crewmate's abductor, commands are given and flags are burned.