I’m not that sure where Asian-American originated. There are no truly large concentrations of peoples whose ancestors were Chinese, Japanese, Korean, Thai, Viet, etc. living in the Great Lakes region, so any language developments regarding those folks tend to trickle back at different times from the West Coast and I have never heard the origins of the “Oriental” vs “Asian” debates.
African-American has a clear (and well-intentioned, but flawed) origin. After a meeting of a number of black social and political leaders in Chicago a few years ago, Jesse Jackson announced that the people whose ancestors were imported to the U.S. as slaves would prefer to be known as African-Americans rather than as blacks or Afro-Americans or any of several other terms.
The intent was to define the black community in terms that were more like the terms used by other ethnic groups. In Chicago (Jackson’s home town) and other rust belt cities (Milwaukee, Detroit, Cleveland, Buffalo, Pittsburgh) from which most of the participants came, it is fairly common for people of various ethnic heritages to identify themselves as hyphenated Americans, forming clubs and mutual aid societies that use the terms Irish-American, German-American, Polish-American, Slovenian-American, Italian-American, etc. Jackson and the other members of the conference felt that, in the midst of these other ethnic groups, labeling their group as “black” made it stand out as different. Hoping to be more like their neighbors, they chose a term that was more like their neighbors’.
Unfortunately, the rust belt, while heavily populated, is not the whole of the U.S. People in the deep South, people in the West, and even people in the outlying farming regions of the rust belt states rarely encounter the clubs of the hyphenated Americans. When he announced his “name change” nationally (and when the various TV networks, newspapers, and wire services deferred to his perceived role as a leader in the black community and began using the term), all the people who were not familiar with the hyphenated American experience immediately saw the term as either divisive or as putting on airs. (The situation has not been helped by idiots in the news media who have used the term as a substitute for “black” and referred to (for example) Nelson Mandela as an “African-American.”)
What started out as an attempt to make the black community more like their neighbors has backfired, making them appear, to a great many people, as less like their neighbors.
Interestingly, polls within the black community never showed a majority acceptance for the term. At one time there was a 53% to 46% preference for the term “black” over the term “African-American”; the most recent polls show that the numbers are now closer to 60% to 39% favoring “black” over “African-American.” (There are always a few people out there who prefer the briefly considered term “Afro-American” from the 1960s.)