No, it is not quite the same as shutting down.
When you click on the Shut Down item in the Start menu, it brings up a window. That window has a dropdown menu where you can select Log Off, Shut Down, Restart, Standby, or Hibernate. Standby is sleep mode and Hibernate is hibernate mode.
Hibernate mode saves the state of Windows in a temporary file and then shuts the computer down. When you start the computer by pressing the power button, it looks in that file and puts Windows in the exact same state it was in when you hibernated. All of the programs that you had open will be open at the same place they were before. Starting from hibernate is faster than a normal start.
So hibernate ends up being just like Sleep except that the computer is completely shut off (if you were to unplug the computer while it is off, it would not ruin the hibernation whereas it would ruin a sleep) during hibernation, starting from hibernation requires you to press the power button, and startup takes longer than the startup from Sleep.
In Power Options of the Control panel you can specify what you want to happen after your computer has set idle for a while. You can choose to have it sleep, hibernate, or a few other options.