On a Server 2012 R2 Essentials machine, I set up the SMTP service with a smart host to relay messages. The setup was identical to a working setup at another location. I could see in the SMTP logs that the server was accepting the messages, and they were accumulating in the Queue folder, but the logs did not show any attempt to make outbound connections.
Finally I looked more closely at the Delivery > Outbound Connections dialog. Somehow Limit number of connections to got set to 0. I set it back to the default of 1000 and mail started flowing out.