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.
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.