The game isn't finished yet, so there are not that many consequences for being a mutant. Right now, some NPCs will refuse to talk to you and mutant hunters will come after you.
In future versions, playing as a mutant will be quite harder. Not only will more people attack you or refuse to talk with you, but there will be some health related issues you will have to deal with.
Right now though, you can enjoy those traits as free points, if you know what you are doing and get the church robes early.