In the UFC, they introduce fighters by their style of fighting. what is the difference between?

Eric K nailed it.

Commonly "Freestyle Fighter" means they started with no real MA training. Basic training that proved them well. Usually moving forward they DO start to train in a few arts.

Versus "MMA Fighter" usually means they have either trained in multiple arts, or strait MMA classes (which teach alot of KB, Boxing for standup, and wrestling and BJJ, judo, for ground).

Hope this helps

Freestyle is also a style that mma fighters use as a style. Like forrest Griffin and rich franklin.