There will be gradual changes to the broader civilization as we move away from the religious ideas of biblical Creationism and toward scientific ideas of Evolution.
The Bible has a story of creation that you can believe literally, or allegorically. Either way it has the power to influence our behaviour. To begin with, it tells us that Men are the dominant sex, and that women were an afterthought, taken from man. Also, man was created by God in the image of God, once again giving us the idea that men are more important than women.
Another aspect of the creation story is that Man is different from all other animals. In the the story of creation, only man was made in the image of God. All the other creatures were crafted to look different from God. We are also to understand that Man's morality comes from God, as told in the story of Adam and Eve. Furthermore we are told that man actually has no innate morality, and is born a natural sinner who needs to obey God, or will suffer punishment.
There are a few other ramifications to this religious story. It becomes easy to believe that God prefers men to women, and it is also a small step further to believe that one type of man is preferred over other types of men. For example, white men over black men.* And there is also a very strong support that whoever believes in the "true" Biblical account has the support of God, and anyone of a different faith should be converted or enslaved or killed.
* Although it is not specifically stated in the Bible, white people seem to believe that God is also white. I can't really prove that of course, but just how many movies, paintings, drawings, have to be made before it's quite clear?
Now what happens to a civilization that is based on such a story of creation, when it finds out that humans were not directly created by God? That maybe we are not so different from the animals. Does it mean that now we have no morals any more? That there is no punishment for being bad? That our culture and race have no claim to a god-given superiority over any other culture, race, or religion? That we can't even insist that men are superior to women?
Some people, without fear of a magical all-seeing being overhead, may get a little (or a lot) crazy. For the vast majority of people, though, I think we will gradually find out that is was not really religion that was stopping them from becoming mass murderers, it was something else built deep inside the human brain. You can say God put it there, if you wish,or that it evolved that way if you are more scientific.
If you need some proof of this, you should look at animals more closely. You will notice that animals are capable of kindness. But you do have to look carefully, as animals, of course do not have exactly the same sense of morality as humans. But nature is full of heartwarming stories of animals doing good. And human history has enough examples of religiously motivated people doing unspeakable evil.
Can we predict what will happen to individuals and societies as these scientific ideas spread? I suspect that we may indeed have more killings and bad behaviours, but I think it is tied more to increasing populations, and new technologies facilitating mass murder, and greater access to information. I suspect that there is not much real difference in the amount of bad behaviour today or in the future, from what there was a thousand years ago. No matter what desperate religious conservatives have to say on that subject. (for example blaming Darwin and the evolution of species for the Nazi holocaust)
But as these ideas of science spread, we will probably find that there will be less religious conflict in the future. Much more freedom of religion and free thinking. More equality between women and men. Less racism. More kindness to animals, and possibly more care for the natural environment. Fewer missionaries trying to convert people. And not a whole lot of difference in average levels of cruelty and violence in society- because it seems more and more apparent, that good and kind behaviour never was a function of religion alone.