Send mail from WP via SMTP. It simply does not work

Hosting:
Godaddy Pro cPanel Linux Unlimited

WP install: Multisite

Email provider: Google Apps (Gmail)

I have used several SMTP plugins and the test always fails. Error:

2015-10-05 21:51:20 SMTP ERROR: Failed to connect to server: Connection refused (111)
2015-10-05 21:51:20 SMTP connect() failed. https://github.com/PHPMailer/PHPMailer/wiki/Troubleshooting

1) Why is this?
2) Kindly reply with a plugin and setting details that actually work. Access Granted. Feel free to test it using a test gmail account informing me of the plugin name used and what I need to do to enter my own gmail account.

Thanks in advance