My reason :
I used to like reading manga when i was a kid (and dont have access to internet) and i like when the author draw funny faces on the character when they mad, laughing, etc. So when i found out that i can see it in anime, it makes me really excited.
Besides graphic, storyline is very important too. Some anime has a very good storyline that can be compared to a real drama series or movie (or even better in my opinion). No drama series can make me cry like when i watched Clannad. I tried to watch a real drama series, but none of them can make me feel attached or feel how's the main cast feeling.
Other aspect that's very important is the soundtrack. Lots of anime has a very good soundtrack. Some anime even become popular because of it' soundtrack.
I also has other reason but i can't explain it since it's personal.
What's the difference between anime and other animated media :
I think anime has more storyline, better drawing, and music compared to other animated media (or maybe it isn't better, but people have different taste and i prefer anime's drawing style). To be honest, i find American's comic / cartoon drawing is odd.