Do you have separate defiinitions that appear to be in conflict? I've heard of the first one, finite state machine, and I think I know what it is. I'm not familiar with the second term although I could probably draw some inferences.
Finite memory and finite state contrast with a machine that has a "tape" which is infinitely long in both directions and thus is not limited in any way in the storage and processing of information. This is also known as a Turing Machine after it's inventor Alan Turing (ca. 1936)