Hello, in their basic meaning they are the same most of the time. But register is far more formal and is often used in connection with official or governmental things like 'to register to vote' or 'to register with the local authorities'. To sign up is more informal and has an element of freedom to it. It means more 'to join'. Example: Please sign up to help at the library this weekend! or Have you signed up with the bird-watching club yet?