What's the difference between "shutting down the computer" vs "shutting down the hard disk"?

I was wondering: What's the difference between shutting down the computer and shutting down the hard disk?

Aren't they the same thing?

That is: I can't use the computer if the hard disk is shut down. If I shut down the computer, isn't the hard disk shut down along with it too?

"Turn Off Hard Disk", is poorly worded in the Windows 7 power properties, the drive is put into "standby", not actually turned off. The drive is "spun-down" to save energy and wear on the motor bearings. On supporting laptops it is put in an even lower standby state. Like ranon said, it occurs when the system or a controller/driver deems that the user is idle for disk activity. It is an independent item, for example the hard drive can go to sleep without the screen saver or screen-off occurring.

If the drive is accessed, the drive wakes back up, The motor spins the platters back up to speed, and then any access to it is returned. You can tell it went to sleep by hearing it spin back up, or because of the seconds of delay you get when trying to do something.

Hibernate is to save the entire present state of operation by moving the ram to disk, and putting the computer into the "off" state.

A computers main Ram needs constant power to keep the Data in it. By shifting the contents of the ram to disk, the contents can be returned to ram. That way even though the ram was empty, the state that the computer was in can be returned mostly. Returning from Hibernate, the contents of the ram are put back in from the hard drive.

"turning off the hard disk" means turning it off while your computer is running, but there hasn't been any hard disk access for some time.

Hibernate stores your main memory's contents and cpu state to hard disk and then shuts down your computer completely.