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. Msql.pm is the elder species. It was written before
DBI was available but it was inspired by an early draft of the DBI
specification. As it has been circulating longer it is more mature and
pretty stable. It's also more complete than DBD::mysql.

Some day in the future DBD::mysql will probably be the clearly superior
competitor in this package. Today you have to decide on your own about
the trade-offs.