Just in case I need to use PHPMailer again...

The important line was $mail->SMTPAuth = true.

Thanks to:

http://www.fasthostinghelp.com/send-mail-using-phpmailer-t98.html

$mail = new PHPMailer(); 

$mail->IsSMTP(); // send via SMTP 
$mail->Host = "mail.domain name"; // SMTP servers 
$mail->SMTPAuth = true; // turn on SMTP authentication 
$mail->Username = "Domain Email account"; // SMTP username 
$mail->Password = "password of an email account"; // SMTP password 

$mail->From = "From email account – from above mentioned domain"; 
$mail->FromName = "From Name here"; 
$mail->AddAddress("To Email account here ","Display name"); 
$mail->AddReplyTo("Reply to email account here ","Details"); 

$mail->WordWrap = 50; // set word wrap 

$mail->IsHTML(true); // send as HTML 

$mail->Subject = "Subject Line of the email"; 
$mail->Body = "Body part of the message – email details</b>"; 
$mail->AltBody = "Alternate part of the body"; 

if(!$mail->Send()) 
{ 
echo "Message did not sent <p>"; 
echo "Mailer Error: " . $mail->ErrorInfo; 
exit; 
} 

echo "Message has been sent"; 

?>