Registering a business name does not make you the owner of the name, it simply shows people that you are operating your business under that name. Additionally, typically this is done at the County or State level which is very different from filing as a US, EU, or country-specific trademark.
Conversely, a trademark gives you an exclusive right to use the name for your scope of business services or products. This means that if others attempt to use your name, or a name that is similar to your trademark, you can stop them.