There are many popular cases for the failure of SMTP connection in PHPMailer and lack of SSL is one of that too.
There might be a case, that the Open SSL extension is not enabled in your php.ini which is creating the connection problem.
So, once you enable the extension=php_openssl.dll in the ini file.
Enable debug output, so that you can really see that SSL is the actual problem or not. PHPMailer gives a functionality by which you can get detailed logs of the SMTP connection.
You can enable this functionality by including the following code in your script;
$mail->SMTPDebug = 2;
By setting the value of SMTPDebug property to 2, you will be actually getting both server and client level transcripts.
For more details on the other parameter values, please refer the official PHPMailer Wiki.
In case you are using Godaddy hosting, then just enabling SSL might not fix your problem. Because there are other serious challenges with Godaddy which you can read here.