Men are usually the travelers, women tend to be more of a nesting creature.
Both sexes have their shallow attitudes at times.
Men would be sent to war more oft, of course, like someone mentioned earlier.
There would be slightly more peace, since women are usually more patient than are men.
I'd say the women would have to largely supress the sex drives of men and their want for power/dominance and egos, alike. Of course, maybe women had a powerful drive of something, in their hidden desires, as overtime, it was suppressed, and became unknown to mankind, those desires in men today may diminish, and 'men' may become something totally other than they are today?
Society would be more feministic and hard driven in acceping and recognizing women first before men, I presume.
A matriarchal society would probably yield different levels of innovation in the world, all together.
But one thing is certain, humans need: food, water, air and love. They must work and work together.
Some of the reasons in "WHY NATIONS GO TO WAR," a book, will probably not change much, the title speaks for itself, such things as: resources, religion, ideologies, etc.
In total, I believe there are seven or thirteen primary reasons categorized by the author scholar.
Happiness may still be a serious problem, as women and men are both pretty materialistic in their own kinds of things. But men are usually more visual and touchy, where women are more observing and care-touchy. I'm guessing too, that more "wussy men" and nerds will exist. Bisexual behavior would probably increase dramatically compared to today's society.
If women were given the same sex drive as men comparitively, the world would self destruct. Women are more emotionally intimate, so it may teach more men in the world to accept this first before they consider their sexual emotions. In love-making and reproduction, I see when women are in control, it may affect pleasure and the relationship, because marriages are a two-way street, but when men ask women for permission too much, it really irritates the female in today's society, however when it comes to this tops- turvy world w/women in control first, men may be more irritated when the women ask them for approval all the time.
As far as government and economy goes, I see rebellious organizations going against different cultures and religions, alike, similar to today's world, bands of people that disagree with the status quo - there will always be that, for some reason, in how people just do not get along with everyone else, as long as there is at least one broken home in the world.
Divorce rates would be significantly less. As men crave more for variety, women usually will stick with their sole "soul mate." The transition from younghood to adulthood, since women mature faster than men, men will be seen as inferior in mentality and not be trusted as much. And when men physically, become equal or greater to women, they will have to go through some kind of societal training to kill their ego/pride that is spawned in the mind.
I see men more regulated then women, almost becoming robots. Like controlled lightning, where if the box is opened, it crash and burns all over the world. Human rights movements to get men upto standard with women, yeah, of course. Worse case scenarios, they'll probably kill the older men that want to teach contrary to a matriarchal society, and rear from childhood, that women 'must absolutely' come first before anything, 'anything.'
Domestic violence would probably be decreased severely. And if it does exist, women would have a system setup where if men became out of control, women themselves, or the matriarchally loyal men would quite the rebellious men or teach them to submit, kind of like a 'pet dog?', if I must so give an example.
In conversation, women are more prone to talk about anything and everything with people closest to them. Secrets would still exist, just on a different level for the sexes.