Comparing "predicted probabilities" and "odds ratios"

Those who use the logistic as well as the margin commands (and both are quite common in ELS) sometimes confront the question of whether to report out odds ratios (from the logistic command) or the predicted probabilities (from the margin command). As one commentator notes in this Stata list discussion (here), "An odds [ratio] is related but not the same as a probability. They are related in the sense that both are ways of quantifying how likely an event (success) is. They differ in that the "odds [ratio] is the average number of success per failure. The probability is the average number of successes per trial." What is the more intuitive of the two will, of course, vary across people.