The R-circle, seen at right, is available when a company has a federally registered trademark. It is required to provide notice to any potential infringers that the mark is federally registered. If the mark holder does not provide this notice, he or she probably cannot obtain profits or damages in a later infringement action.
The "TM" is used for pending or unregistered trademarks. If you filed a federal trademark registration, but it is not officially registered yet, you can use "TM." If you have an unregistered mark, but claim rights in it, you can use the "TM."
The difference is easily illustrated with sports logos. When teams introduce a new logo, you will see a "TM" next to the mark. This means that the team has not used the mark long enough for a federal registration to issue. For instance, take a look at the new University of California-Berkeley logo. If you notice the top image on the link, it has a "TM" next to it. This logo was just unveiled a few weeks ago. If you take a look at the logo in a few years, you will notice that it has changed from a "TM" to an R-circle.
For another example, take a look at one of the University of Nebraska's logos. This logo has been around for a while, and thus has a federal registration.