Why is it more acceptable for a man to date a woman significantly younger
than him than it is for a woman to do the same?
It's a cultural legacy from the time when men and only men were primary bread-winners. Because women couldn't work and because they were often legally prohibited from inheriting property, fathers were economically encouraged to marry off their daughters as soon as possible (think of Pride and Prejudice). At the same time, men could not afford a wife until they had established themselves in the world (in business, the trades, etc). These twin imperatives created the spousal age disparities of which you speak, and which are still commonplace in traditional cultures in the Middle East, parts of Asia, etc.
In post-industrial economies where women have entered the workforce and achieved a measure of economic independence, it is not unusual for such women to marry much later, or to not marry at all. Many of my women friends are careerists, in their mid-40s, who have never been married. This is increasingly becoming the norm, especially for those women who have the opportunity to attend elite colleges. The stereotypical "cougar" is precisely this sort of highly educated and super-successful career-oriented woman.
In short, economic conditions largely determine what is or is not acceptable. But these conditions can change, and we're seeing those changes in effect now, at least in the post-industrial West.