I've looked into this, and it looks like GoDaddy is blocking outbound email (by blocking outbound network traffic destined for common SMTP ports). When I called GoDaddy support to get it fixed, the support rep with whom I spoke claimed that they were not blocking outgoing mail. But based on my testing via SSH, the rep is either lying or just plain dumb -- the outbound traffic is obviously being blocked.
I need to call GoDaddy support again and try to get a better support rep. I'll see if I can do that tomorrow. Stay tuned...