How does do not disturb in iOS 6 differ from the physical silent switch?
Do Not Disturb stops all notifications from occurring (so your screen won't light up). Silent will still cause the screen to light up, vibrate if enabled etc. The notifications will still be shown in notification centre.

If you receive a call while on do not disturb the caller gets a call failure and you will be shown a missed call on the recent list as opposed to silent where it will still ring (screen change etc) and you can choose to answer.

Do not disturb will allow certain selected calls to come through (you can customise this in notifications in settings). It can also be scheduled which is what I have been using most.

There is a section on Apple's iOS 6 page but it is not great.