Is a Markov chain the same as a finite state machine?

Markov chains can be represented by finite state machines. The idea is that a Markov chain describes a process in which the transition to a state at time t+1 depends only on the state at time t. The main thing to keep in mind is that the transitions in a Markov chain are probabilistic rather than deterministic, which means that you can't always say with perfect certainty what will happen at time t+1.

The Wikipedia articles on Finite-state machines has a subsection on Finite Markov-chain processes, I'd recommend reading that for more information. Also, the Wikipedia article on Markov chains has a brief sentence describing the use of finite state machines in representing a Markov chain. That states:

A finite state machine can be used as a representation of a Markov chain. Assuming a sequence of independent and identically distributed input signals (for example, symbols from a binary alphabet chosen by coin tosses), if the machine is in state y at time n, then the probability that it moves to state x at time n + 1 depends only on the current state.

I believe this should answer your question:

https://en.wikipedia.org/wiki/Probabilistic_automaton

And, you are on to the right idea - they are almost the same, subsets, supersets and modifications depending on what adjective describes the chain or the automaton. Automata typically take an input as well, but I am sure there have been papers utilizing 'Markov-chains' with inputs.

Think gaussian distribution vs. normal distribution - same ideas different fields. Automata belong to computer science, Markov belongs to probability and statistics.

The two are similar, but the other explanations here are slightly wrong. Only FINITE Markov chains can be represented by a FSM. Markov chains allow for an infinite state space. As it was pointed out, the transitions of a Markov chain are described by probabilities, but it is also important to mention that the transition probabilities can only depend on the current state. Without this restriction, it would be called a "discrete time stochastic process".

Whilst a Markov chain is a finite state machine, it is distinguished by its transitions being stochastic, i.e. random, and described by probabilities.