connect access denied in perl

[Jie Tang]

the user "mysql" is given all priviliges,
another user is given db table privileges (INSERT, SELECT, etc.),
the database files are owned by user mysql,

in the same perl DBI script, at the very biginning, the line:
$dbh = DBI->connect($dsn, $user_name, $password, {RaiseError => 1});

works only when the $user_name = "mysql", 

the other user is denied access to the database.

does anyone know why?

Thanks!