Why do people care whether free will exists?
Well, the answer to this question really matters.
First, it goes about personal responsibility.
If you cannot choose, you cannot be hold responsible for your choice. Belief in the nonexistence of free will somehow frees you morally and you become more likely to act in a less morally responsible way (the point is supported by relevant experiments). This is why many experts do instinct that the fact that free will is just a brain mechanism should be kept a secret.
Second, the human society – juridical system in particular – is based on the belief that a person can choose whether to commit a crime or not.
Whether free will exists or not determines whether people blame those who offend, bully, cheat, etc. or take their actions as inevitable and not dependent on them. Some experts do encourage the spread of the latter view so that the society becomes more responsible for eliminating social disadvantages which eventually result in poor choices of some individuals. It sounds like utopia/anti-utopia for me though.
Third, the resolution of this question matters for religious people. Christian religion suggests that a person can choose to pursue the path of evil or to listen to the word of God and live by it. People assume personal responsibility for their sins and it determines where you go after death. If people cannot choose and their genes+upbringing makes them doomed, it changes it all…
You can find evidence to all the arguments in the literature review on the point – the post Free will does not exist: Better keep this a secret? in my blog.
You will also read there about “a third way” in the understanding of free will, developed by Stephen Cane. He suggests a very sound theory I personally support, which reconciles brain research and the idea of personal responsibility. Would love to hear your opinions about it in the comments to the post!