To shut down, hold the top, screen lock button. A slider will appear letting you shut down the iDevice.
To reset, hold down the top, screen lock button and the home button simultaneously.
You might need to reset if shutting down and restarting doesn't fix a problem you're troubleshooting, or if there's something preventing normal shut down.
Bruce Blakely and Colleen Thompson both questioned my claim that a reset can lead to data loss.
I can only assume they didn't do any testing, because a very simple test (described below) proves my point.
But first I need to explain the difference between a reset and a proper reboot.
The easiest way to explain this is by comparing it to a Mac. On a Mac, when you select "Shut Down" on a Mac, the system notifies all running applications that a shut down has been requested. This allows those apps to write any unsaved data to disk and close open files. The system also saves cached data to your drive. If you have unsaved changes in a document, you'll be prompted to save or discard those changes.
You can also power down a Mac by holding the power key for 5 seconds. This simply cuts power without alerting applications. Unsaved documents are lost, as is cached data not yet written to the drive.
On an iOS device, using "slide to power off" is the equivalent of the Mac's Shut Down command, notifying apps that the device is about to be turned off. And a reset is like a forced shutdown.
Now, here's the proof. Create a new note using the Notes application. Type in a few words, then use "slide to power off" to shut off the device, then turn it back on. When you return to the Notes app, you'll see that your note was saved.
Now try the same experiment, only do a reset. When your device reboots and you return to the Notes app, your note is lost.