An understanding of finite-state machines is useful, because that way you'll understand the limitations you have when using regular expressions, which have been grossly misused.
It also helps you write optimal regular expressions - for instance if your regular expression clearly describes a DFA, then it will probably perform well. Otherwise in many cases you can optimize it - just as you can turn any NFA into a DFA.
It isn't so clear and cut these days however, as the regexp capabilities provided by modern libraries exceed the capabilities of finite state machines - for instance the engine in Perl 5.10 has support for recursion, which means a regular expression in Perl 5.10 can describe a Pushdown Automaton (PDA) instead, so things get way more complicated.
Either way, I suggest you do some reading on Automata Theory, because it's fun and not that hard.