Does it matter whether we believe in free will or not?

From Genschow et al.:

Significance

The question whether free will exists or not has been a matter of debate in philosophy for centuries. Recently, researchers claimed that free will is nothing more than a myth. Although the validity of this claim is debatable, it attracted much attention in the general public. This raises the crucial question whether it matters if people believe in free will or not. In six studies, we tested whether believing in free will is related to the correspondence bias—that is, people’s automatic tendency to overestimate the influence of internal as compared to external factors when interpreting others’ behavior. Overall, we demonstrate that believing in free will increases the correspondence bias and predicts prescribed punishment and reward behavior.

Abstract

Free will is a cornerstone of our society, and psychological research demonstrates that questioning its existence impacts social behavior. In six studies, we tested whether believing in free will is related to the correspondence bias, which reflects people’s automatic tendency to overestimate the influence of internal as compared to external factors when interpreting others’ behavior. All studies demonstrate a positive relationship between the strength of the belief in free will and the correspondence bias. Moreover, in two experimental studies, we showed that weakening participants’ belief in free will leads to a reduction of the correspondence bias. Finally, the last study demonstrates that believing in free will predicts prescribed punishment and reward behavior, and that this relation is mediated by the correspondence bias. Overall, these studies show that believing in free will impacts fundamental social-cognitive processes that are involved in the understanding of others’ behavior.

Also, you should have a look at Frith's essay on how our illusory sense of agency has a deeply important social purpose. Belief in free will and agency is important if a distinction critical to all legal systems is to be made: between intentional and accidental wrongs. Further,

Responsibility... is the real currency of conscious experience. In turn, it is also the bedrock of culture. Humans are social animals, but we’d be unable to cooperate or get along in communities if we couldn’t agree on the kinds of creatures we are and the sort of world we inhabit. It’s only by reflecting, sharing and accounting for our experiences that we can find such common ground. To date, the scientific method is the most advanced cognitive technology we’ve developed for honing the accuracy of our consensus – a method involving continuous experimentation, discussion and replication.