So what happens if you put a sleep statement at the end of the loop? Something like:

#!/usr/bin/perl
use Net::SMTP;
use strict;
use warnings;
for(1..2) {
  my $smtp = Net::SMTP->new('smtpauth.earthlink.net', 
                          Hello => 'milkbone.org', Debug => 1) or die 
+"death";
  $smtp->auth('batkins86@earthlink.net', '***************************'
+);
  $smtp->mail('monitor@milkbone.org');
  $smtp->to('test@batkins.com');
  $smtp->data();
  $smtp->datasend("To: postmaster\n");
  $smtp->datasend("\n");
  $smtp->datasend("A simple test message\n");
  $smtp->dataend();
  $smtp->quit;
  sleep(5);
}
[download]

Perhaps Net::SMTP hasn't fully cleaned up before you call it again? Don't think it would matter but I'm trying to cover all the bases...