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.
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.