Difference between a turing machine and a finite state machine?
The major distinction between how DFAs (Deterministic Finite Automaton) and TMs work is in terms of how they use memory. 

Intuitively, DFAs have no "scratch" memory at all; the configuration of a DFA is entirely accounted for by the state in which it currently finds itself, and its current progress in reading the input.

Intuitively, TMs have a "scratch" memory in the form of tape; the configuration of a TM consists both of its current state and the current contents of the tape, which the TM may change as it executes.

A DFA may be thought of as a TM that neither changes any tape symbols nor moves the head to the left. These restrictions make it impossible to recognize certain languages which can be accepted by TMs.

Note that I use the term "DFA" rather than "FSM", since, technically, I'd consider a TM to be a finite-state machine, since TMs by definition have a finite number of states. The difference between DFAs and TMs is in the number of configurations, which is the same as the number of states for a DFA, but is infinitely great for a TM.