If you register an LLC then no one else can use that name in the same state in which it is registered.
Corporate entities are state creations. There isn't a federal registry of corporations.
Trademark is separate. You can register a corporate entity using a name, and someone else can use that same name as a trademark.
Two different companies can use the same trademark, like Delta airlines and Delta faucets.
Trademarks are associated with a particular of goods/services and often completely different from the name of the company that makes the product and owns the trademark.
Some companies do use their corporate name as a trademark.
If you want to protect the name as a trademark, it's best to have it registered with the Federal gov. (US Patent & Trademark Office).
However, just because a state let you register a name for a corporate entity doesn't mean you'll be able to register that same name as a trademark. Befoire you apply to register a mark, it's a good idea to have a search done. The Trademark office will refuse to register marks that a similar to existing registered marks. So even though your mark may be literally different (different spelling, additional words) if the trademark examiner thinks another registered mark could be confused by the public as being the same, they will disallow the application.
The "R" and "TM" are legally different. The circled R means it is registered. You are not allowed to use that if it is not registered. "TM" can be used for non-registered marks to put people on notice that you are using it as a trademark, but the mark is not registered. This gets a little confusing because there are common law rights to marks that are not registered.