There is no advantage in the code you posted. In your code, using the delegate just adds complexity as well as an extra runtime cost - so you're better off just calling the method directly.
However, delegates have many uses. "Passing around" to other methods is the primary usage, though storing a function and using it later is also very useful.
LINQ is built on top of this concept entirely. When you do:
var results = myCollection.Where(item => item == "Foo");
You're passing a delegate (defined as a lambda: item => item == "Foo"
) to the Where
function in the LINQ libraries. This is what makes it work properly.