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 also used when you want to delay the start of a service to allow other services to start completely, such as the SCCM Client, which does a delayed start to allow the WMI service to start fully as it is dependent on it being fully up and running.
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.