A Trading as name is a name chosen by a business that is different from their registered company name. Unlike the registered company name, the trading name is never officially registered with Companies House.
It is the name a business has chosen to call itself, usually because it is more marketable than the registered company name.
You are free to trade under it, but you cannot solely be known by your trading name, as you must display your registered business name of signs, receipts, and official documentation.
A trademark is something that identifies the products or services of a specific business.
It is a valuable business asset that needs to be protected to prevent others from trying to use it to sell their products and services.