Oddly enough the only known mammals that bleed on a cycle are not all on the same evolutionary tree: bats, some primates, elephant shrews and one species of mouse are known to menstruate. It makes menstruation all the more puzzling to me. Sadly we can't ask any of the other mammals about whether they experience menstrual pain, but I imagine they do.

I am no doctor, but I have read that during menstruation the human cervix dilates to 10 percent the size it becomes during childbirth. If you've ever had a migraine go beyond the ocular stages you'll know this kind of pain and what causes it; during the first phases of migraine the blood vessels constrict, causing visual distortions. In some people, once they release from the constriction phase, they do not stop on their own and continue expanding, causing excruciating pain that can be comparable to menstrual pains.

Main difference being that the former is more of a "lay still in silent darkness for hours with a cold pack" type situation and the latter is a "writhe around in pain silently while you go about the next handful of days chugging teas" kind of pain.