The ® implies that the mark was registered with the US Patent and Trademark Office. The ™ denotes a common law trademark.
In the US, federal trademark registration grants you extra protections. Namely the right to use the name, barring any prior use, across the entire United States. It also puts other people on notice that you own the mark. That way if someone infringes, there is the presumption that the infringement was not innocent and can lead to increased damages. There are other advantages that accrue with time that prevent others from challenging the validity of your mark as well. You will also have access to the Federal Court system, instead of the state systems.
However, a common law trademark provides you protection in the geographic area that you use the mark. It does not mean that anyone else can use your mark. They both grant you the right to use your mark to the exclusion of others in your business area and, depending on the mark, geographic region.
If you simply use TM, you may have very little legal recourse if it is infringed on by another. This varies by country as to how much, if any, legal recourse you may have. A registered trademark, i.e., ®, entitles you to legal protections and you'll be able to send those nasty cease and desist letters from unkind attorneys to those you infringe on you and the law backs you up. The registered trademark costs some money, a few hundred if I recall correctly, and it takes some time to acquire.