From what I understand, Apache::DBI provides certain performance benefits over DBI in much the same manner that mod_perl does over standalone Perl, but isn't required to use DBI in a mod_perl environment.
Essentially it makes your database connections persistent.
So when you establish a connection at the start of a particular mod_perl script it intercepts the request and returns a pre-connected DB handle. Similarly it intercepts the disconnect request at the end of the script and leaves the connection to the DB open ready for next time it's needed.
For DB's that are slow when establishing connections it can make a considerable speed difference.
Carl