There are at least 2 reasons. One is precaution, animals are vulnerable doing childbirth, screaming attracts unwanted attention. Not all people scream anyway. I have been near several births in remote areas in Bolivia and Honduras, the women didn't scream. SO, part of the screaming is custom.
Human babies have tremendously big heads compared to the size of the mother and her birth canal. Usually once the head is delivered the pain starts dropping immediately. Other animals do not have the problem of over-sized heads, the stress and pain is less.
Another part that may be important is that modern births are done in an unnatural position. This may add to the pain. At least it seems to prolong it.