Who will say “Age before beauty” more often? Man or woman, young or old?

I think this definition has it about right. The expression started in the mid to late 1800s, as a polite way for an older women to acknowledge the courtesy of a younger woman (in standing aside to let her enter a room first, for example). But it soon became a gallant thing for an older man to say to a girl, and to be used jokingly between other pairs.

Effectively, it's a long-surviving catchphrase (sense:2 there), which doesn't really "mean" anything beyond the literal. No doubt it's been said often by twins - who themselves invariably know which one is a few seconds or minutes older. On different occasions either twin might say it. The sex of each twin being irrelevant - the only requirement of the context is the older one goes or gets first.

It's normally said jokingly these days. It would be a bit "cheesy" to use it in contexts where any element of latter-day gallantry was intended.

Whenever I've heard this term in contemporary use, it was meant to be a good-natured backhanded compliment, often between a worker and his supervisor (I've heard this most often in the work setting, as it seems to work best with people who know each other and are on good terms).

Let's say Tom (age 28) works for Jerry (age 44), and the two of them are waiting for an elevator. When the elevator doors open, Tom gestures kindly, as if to say "after you," but as Jerry takes the first step, Tom says with a smile, "Age before beauty."

The implication is, "Not only are you older than me, but I'm better looking, too!" (After all, if it was beauty before age, Tom would have stepped on first – at least, that's what he's implying).

The UK site Phrase Finder mentions that the idiom is usually uttered "by the older person in order to flatter the younger," but in my experience in the U.S., it's usually the other way around – the younger says it to the older as a way to tease him or her about getting old.

As an aside, FumbleFingers mentioned the irony of twins using this phrase, because their age difference is mere minutes. That reminds me of one of the funniest utterences I've ever heard: a girl who was told, "Well, you're so ugly, who would ever want to marry you?" by her identical twin sister!

Back in days of old, it was considered common courtesy for a man to enter or exit a doorway or stairwell or room first. This courtesy held the intent that the elder man would inadvertantly ensure the woman's safety by being first to enter a new territory where some or other danger might be unforeseeably present. As time grew on and people became less paranoid and suspicious, it became courteous for a younger person to give way to an older person, out of a sign of respect. For an older man to allow a woman to go first, irrespective of age, was considered an act of chivalry. Nowadays the phrase is used more often as a quip by the younger to emphasize the elder persons age in contrast to their own youth and beauty.