Two years ago, I noted that a survey of black Americans reported a tie between those who prefer "black" and those who prefer "African-American." I just ran across a Gallup survey (from last Summer) that reports a slightly more complex picture: The survey (again, of black Americans) asked,
Some people say the term "African-American" should be used instead of the word "black." Which term do you prefer -- "African-American" or "black," or does it not matter to you either way?
61% said it doesn't matter, 24% preferred African-American, and 13% preferred black. The maximum margin of error (at, I take it, the 95% confidence level) is reported to be 6%, but as I understand it the margins of error in such surveys are generally lower for the smaller percentages, so the 24%/13% difference is statistically significant at the 95% level (though you shouldn't take the precise magnitude of the difference to the bank).
My conclusion: Even if you believe that members of various identity groups should generally be called by whatever term the group prefers (I've criticized that position here, but let's assume it for the sake of argument), in this instance we see no clear preference. You might want to use "African-American" to maximize the appeal to black audiences, or for other reasons -- or you might want to use "black" for other reasons as well. But I don't think you can be faulted as a matter of good manners for using a label ("black") that three-quarters of the black American population finds to be either no less acceptable than "African-American" or actually better.
And, of course, you should recognize that while the two terms are roughly synonymous in the U.S., they are very different when you're considering the world at large: "Black" is a racial category, and "African-American" (in its common American meaning) is a subset of that category limited to those blacks who are also American. For an example of an error related to this unthinking substitution of "African-American" for "black," see this CNN transcript: "Hard to say because it's been 11 days since two African-American teenagers were killed, electrocuted during a police chase, which prompted all of this" -- "all of this" being rioting in France, which was triggered by the deaths of black teenagers who I'm pretty sure were not American tourists. (Thanks to Wikipedia for the pointer.)