It sounds like your mail server is configured to use POP before SMTP authentication, a kludge which is older than any SMTP authorization methods.

So you have to use Net::POP3 before you use SMTP. That is,

my $pop3 = Net::POP3->new($smtphost) 
   or die "Cannot connect to host $smtphost: $!\n";
$pop3->login('me@me.com', 'password') 
   or die "Cannot login: $pop3->message\n";
$pop3->quit;
undef $pop3
my $smtp = Net::SMTP->new(etc...);
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