Shutdown is just your basic close all running processes, turn off all power, byebye pc.
Hibernate is when you want to preserve the current state of working, but without having a fully powered up PC.
If you hibernate, it will remember which programs you had open and basically continue where you left off.
I suppose you use hibernate when you need to quickly go away from the computer for a while, but dont want the hassle of remembering exactly what you were doing when you come back.