This module offers a diffent set of functionality than DBI. DBI (from what I can see) seems focussed on whole projects that use a database, providing abstractions an such. Net::MySQL on the other hand doesn't require you to model your database at all; it is much more lightweight.
It allows you to provide an arbitrary query string (like you would type into the mysql interpreter) which it will send to the database (either on the localhost or using TCP/IP). It them allows you access to the results. It also is pure Perl, which was appealing.
This module may have fewer uses than DBI, but it fit my need for automating a set of queries I had previously developed (for an existing database) in the mysql interpretter. I wanted this in Perl so I can have more flexibility in the output.