Thanks, bbergman! I'd a similar problem with Godaddy and the CSCF plugin (clean and simple contact form) suddenly stopping working. Installed WP SMTP and all was well.
In my case, the correct host is smtpout.europe.secureserver.net as my WordPress site is actually hosted on Godaddy and I'm in Europe. I think for Godaddy hosting customers elsewhere, it might be smtpout.secureserver.net but they can find this directly from their Godaddy -> Email management -> Email setup page.