The really short answer is that there is no difference conceptually between a Rails fixture and a Factory Girl factory. This is just different syntax to accomplish the same thing.
However, this gets confusing because the terminology of general testing/coding concepts has been co-opted by individual projects in the Rails/Ruby world. In general:
Basically in Rails, both Fixtures and Factories (as defined by Factory Girl), are just ways of creating objects. You are supposed to use them to create your Test Fixture, or the setup state of your test. The confusing part is that you can use multiple Rails "Fixtures" to create the Fixture (baseline) of an individual test.
Factory Girl itself is an implementation of the Object Mother pattern, that describes a way of creating test fixture setup using personas.