A straight up apology is one in which someone does not say, "I apologize if anyone was offended." One apologizes for actions or words simplicter. If done quickly, then, as suggested by authors down through the centuries--including the author of the Hagakure, the book that describes how the samurai ought to act--the offense may seem to vanish. The listeners or readers now have an obligation to accept the apology, or at least seriously consider doing so.
If one waits too long (and this is a necessarily elastic concept of "too long), then the straight up apology may not help at all. We use a complex code of timing and appropriate content.
All this is brought to mind by a written apology offered by an officer of a civic organization I belong to. By writing the apology, he really is committed to its promulgation. The offense may not really involve any ethical or legal matters--thus, he may not have committed an offense at all. But rather than draw out the arguments, he just sent an apology to all the members. I thought that was well done.
