The bottom line is that the JDK regex implementation supports a rich set of regular expressions, whereas dk.brics.automaton supports only the basic features. If you are not concerned about speed - for example, you are just doing a few matches on strings of modest length - then use the JDK implementation. However, if regular expressions are a bottleneck in your program and you don't need advanced regex features, then dk.brics.automaton might be a great fit.
Given the issues of the JRE's size, the question should be asked: does the JRE need a full Perl-like regular expression language, or is a simpler expression language sufficient?