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: 
From my understanding, it's simply a delay before launching the service.

Later versions of Windows introduced this to ensure that they didn't trip over each other's feet during the boot process (having a gazillion processes starting up at the same time is not conducive to performance).

The documentation for this feature states that services marked thus will be started "shortly after boot", hopefully once the boot-time-required services have settled down a little.
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.