Well, it would certainly shake up my morality a bit, which would then change my actions. My ideas of punishment and reward would become more utilitarian as I held people less "responsible" for doing good or bad, for example.

However, if you're asking "what would be different if you'd been living in that universe all along and never found out," I must admit I can't think of anything. Wait, nevermind. "The bell inequalities wouldn't be violated." Or "fermions wouldn't be identical particles." "Arithmetic would be inconsistent." But it's possible to imagine "just so" theories that would fit observations without having much free will. I wouldn't say a Boltzmann brain has free will in the second before it boils away into the plasma.

Still, I think Occam's razor helps rule that stuff out. I'll have to think about it more.