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Actually they are exactly equivalent.

See for example http://en.wikipedia.org/wiki/Regular_expression ("Regular expressions in this sense can express the regular languages, exactly the class of languages accepted by deterministic finite automata.")

Most of the regex engines in languages (like Perl) allow keeping track of state by using back references. If you're talking about this, then theoretically, they aren't "finite state" machines.

FSMs can't handle nested tags. Pushdown Automata can.

Most "regular" expression engines in modern programming languages can handle back-references (e.g. "(.)\1"), which are actually so complex that you need a full-blown Turing machine to handle them.

Nested tags can't be matched by fsms.