What is the difference between finite automata and finite state machines?
As far as I understand, both have "states", and "actions" that make the machine move from one state to another upon an input signal. Thus the conceptual ideas are the same. There is some difference in the details.

In FSM for circuit designs the input signal is mostly assumed to be a bit (binary), whereas in finite state automata one can have a general "abstract" alphabet of input symbols. Second, a FSM also generates an output, associated to the state reached, also binary. In automata terminology this 'extension' is called a Moore machine. Automata however have final (or accepting) states, that signal a favourable input read. Finally, FSM are mostly deterministic, i.e., for each input in a certain state there is one next state. In automata theory one also considers the nondeterministic variant where one might have choice in where to move.