Registering as a sole trader informs HMRC that you are going to be self employed and will be earning money through your own business. It does not register your business, or create a business; it just tells HMRC what you are doing so that they know how to tax you. As a sole trader you are trading as an individual and taking on all financial and legal liability yourself.
Doing a company formation sets up a company as a separate legal entity from yourself. By trading through the company you can benefit from ‘limited liability’ from financial and legal issues. It also means that multiple people can formally take joint ownership of the business by allocating shares in the company