Consider the type of task returned, the first one yields Task<Task<int>> while the second yields Task<int>. So really the first one is a Task representing starting of the inner task, while the second, unwrapped, represents the Task returned by the inner method representing the service starting. Finally you can also Unwrap the first and get the same effect without the async/await which is unnecessary here. None of this really covers what the need for StartNew is at all in this case just reviews the return types your looking at.
Consider the following code:
public class AsyncTesting
{
public void StartServiceTest()
{
Task<Task<int>> tsk1 = Task.Factory.StartNew(() => StartAsync());
Task<int> tsk2 = Task.Factory.StartNew(() => StartAsync()).Unwrap();
Task<int> tsk3 = Task.Factory.StartNew(async () => await StartAsync()).Unwrap();
}
public Task<int> StartAsync() => Task.Delay(2500).ContinueWith(tsk => 1);
}
The method that does not Unwrap returns a Task that represents starting the internal Task not the work it does.