Question: What is the difference between Automatic and Automatic (Delayed Start) for a property setting for a Windows service? i.e. what do I gain or lose by setting my service as one or the other?

Running the service on Windows Server 2008 x64

Answer: 
This is mostly used for services that starts on with AD accounts. If the service is trying to start before the server has network connection to contact the DC, it will fail and sometimes the system gets stuck and gets unreachable.

The delay ensures the service will start once network connection is up in order to use the AD account for it.
Here's an example. I have a SonarQube service that depends on my MySQL database service. So, I set the MySQL service to automatic to get the database up and running when the machine boots. I have the SonarQube service set to the automatic (delayed) start to make sure it starts up AFTER the database service it depends upon is started.

If you have a dependency like this it can be helpful to use the delayed start to make sure dependent services start up in the right order.