Mysql.pm and DBD::mysql implement two different approaches to
communicate with an mysql server. DBD::mysql is built 
upon the DBI, the generic Perl Database Interface. It brings you an
identical interface to a broad variety of databases and is in this
regard comparable to ODBC. The advantage of the DBI approach is
portability and interoperability. Mysql.pm is the elder species.
It was written before DBI was available but inspired by an early
draft of the DBI specification. As has been been circulating longer
it are more mature and pretty stable. It is also more complete
than DBD::mysql.