I don't know if I'd call them a weakness, but when something happens that needs to be dealt with, you can't have people just falling apart and crying on you.

I think it's the intersection of cultural, biological, and evolutionary tendencies, not a single factor. Men are conditioned from childhood with images and ideas of what manhood is: tough, independent, invulnerable. Look at the shows kids watch -- they're full of heroes with super powers who can stop bullets with their hands or fly, etc. You don't see a lot of superheroes crying because of social injustice or whatever. Then there's biology: testosterone tends to augur for aggression. From an evolutionary standpoint, being competitive, aggressive, and strong are favored by selection -- such males are more likely to reproduce and have those traits passed along, etc. All of this adds up to a predisposition to avoid anything which seems like "being out of control", and strong emotions lead that way. The one exception is anger -- which is an aggressive emotion, and tends to feed the sense of being powerful and invulnerable. Pretty much any other emotion is potentially an invalidator of "I'm in control here, I'm in charge, I'm on top". This tendency is only a tendency. Individuals vary widely, and a single individual can transcend his own conditioning and develop a richer and more sophisticated range of emotional experience and expression.

It exposes your emotional buttons to people who might exploit it.