Asian American is a pretty broad term and I don't hear it used in the Asian community, which is huge where I live. I hear people say they are Chinese or Korean or Hmong just as those of European descent would say they are Dutch, French, English, German or Greek.
Here's the difference with the African American thingy. Uh, do we have Malians here? Do we have Ghanese? No, because blacks in this country lost their history when they were brought here as slaves. All they know now is the continent their ancestors came from, thus "African-American."
And if they want to stress the American where those of us of European descent would not bother, perhaps that has something to do with their invisibility in our culture and the pervasiveness of the idea that somehow blacks are somehow "less" American than those of us who are white.
If you want to understand why a black person would care to be called African-American, begin by studying their history, as they tell it.
It's a fascinating story and well worth your time.