I'm afraid you'll never get a good all encompassing answer about what it means to be male. The old truth of larger in group variance than between group variance is true here as well. You'll probably find more similarities between extrovert men and extrovert women than between introverts of the same sex. However with that said there are some things that are more common to men than women so I'll see if I can address the last part: why men don't cry.

This will of course be a major generalization so forgive me for any stereotypes. Most of the male experience versus the female experience comes down to the notion of intrinsic value. Where females provide value just by existing (or by being a potential mate/sexual object depending on your viewpoint) males do not provide any value unless their net contribution to the group is positive. Hence males will strive to reach a position that is seen as valuable to the group or not to burden the group. It is when someone focuses too much on the latter part of not being a burden that we usually start to see some problems associated with being male. Firstly it is not showing emotion (viewed as increasing someone elses emotional labour), then it is not asking for help (viewed as increasing emotional or regular labour) and finally committing suicide (viewed as removing the net cost of their existence). So trying to get a male to show their emotions without addressing how to increase their worth in society at large can be viewed as just increasing the visibility of their lack of value. In a sense it will be like putting their short comings on display at the town square. This will be viewed as a major threat where the logical response is anger in an attempt to deflect criticism and rally resources.

Thus in regular therapy where you can use the 'carrot' of caring about someones emotions as a way to improve the alliance it can for the above type of males become a 'stick'. It can therefore be helpful to turn it around and talk about more instrumental parts of their issues as the 'carrot' while working on building resilience towards the emotional aspects.

I almost didn't post this because I really don't feel that I have the scientific backing for this, but maybe someone else can fill it out, so take it for what it is: an anecdote.