The title says it all. All that's left for me to do is tell you how through the use of shocking anecdotes.
One of my colleagues and I were able to sum up the difference between the genders quite concisely. We approached the divide on the plane of physical appearance, since is this one of the realms in which we humans are best able to understand baboons.
Male baboons are greatly fortunate in being very handsome. They are blessed with thick, flowing coats of hair, square jaws, pristine bodies, and ears which are usually entirely intact. The females have misshapen nipples, scarred tails, mangled finger permanently bent out of shape, and ears which are often so damaged they become our primary means of identification. Furthermore, when a female is frightened, her tail sticks into the air, her lips curl back revealing her yellow teeth and bright pink gums, and she dashes about wailing frantically. It is a hideous sight to behold.
Please do not get the wrong impression - I do not fault the females, for how can I? Most of their unattractiveness is caused by baboon-on-baboon violence, not innate uglyiness. Everyone beats up on female baboons. From a human standpoint, they were dealt a very poor hand, indeed.
The most obvious (to us humans) source of violence is the males. Our minds are swamped with tales of domestic violence in human society, offering us some preparation for what we observe in the troops. Adult male baboons always sit much higher in the absolute hierarchy than females, mostly because of their immense size advantage. This size is exactly what defines much of their relationship with the females. Virtually all adult males, not just the alpha, will chase the females. We call this herding. Males use it as a tactic to control the movements and interactions of females - the unfortunate victim of the herding, specifically, but also any other females in the vicinity.
Most of the time, the chased female will just grimace and shriek horribly while running madly from the male, but the male can pretty much always catch the female if he truly desires. Maybe he's really angry at that female, or displacing some other aggression, but there are times when he catches, them, and its not a pretty site. The effect on the female is primarily mental and emotional, nevertheless the females rarely escape physical harm.
Like their brothers, juveniles will also chase the females around. An adult female is about the same size as a four year old juvenile, and the females are not built for rigorous combat. So, those juveniles can push them around just as easily. Injury is rarely incurred by the females in these conflicts, but I can personally attest that these males tend to be huge assholes, and are a giant nuisance to constantly deal with.
Yet, a female's worst injuries are invariably inflicted by other female baboons.
A female's time is largely consumed with socializing with other females, and the young children. The dominance hierarchy is thus very important to these female baboons, as it affects mate selection, alliances, grooming partners, etc. Basically, baboons hierarchies are rigid. Part of why they are so rigid is because is you step out of line, you'll end up starting a fight. The torn ears, ragged tails, and scratched faces are the obvious signs of the lifelong sources of suffering
Males leap about, wahoo at each other, engaging in ritualized aggression, which is when the fighting doesn't actually come to blows, but one's strength is displayed via alternative methods. In the case of the male baboons, they engage in "wahoo battle" where the male who can run around and shout at the top of his longs the longest is clearly shown to be the strongest. This way, no one gets hurt, which is good for everyone, and the outcome remains the same. From my observations, it appears that females engage in a lot less ritual aggression, and a lot more real aggression
I have seen female baboons gradually tear the skin from the skull bone of a live and struggling female over the course of several days (a tale which may one day be told in full...). She lived, but her scars will not fade anytime soon.
Violence appears to be the worst part of a female baboon's life. Still, when they aren't being abused, they still have to to deal with the joys of pregnancy and subsequent high infant mortality rates. There's no such thing as monkey abstinence, so until they become menopausal, female is either pregnant or suckling some brat who is constantly yanking on their nipples (when I am able to get a good shot of this, I will show you these nipples - they are impressive). They're often still suckling when they begin cycling again, which doesn't stop the males from following them around all day, and copulating on a whim. Immediately following ovulation, I'd guess a female will have to copulate about a hundred times in a day. Plus, they have to carry those big, tender genital swellings around all day.
Link: My opinion piece on recent selection and educational attainment