In iOS 5, users must enter their password whenever they wish to download an app, whether paid or free, and every time they wish to download an update.
In iOS 6, upon pressing the 'Free' button on the App Store, the button changes to "Install App" and another press begins the install process. In another change designed to make the purchasing process quicker and easier, the App Store no longer returns users to the home screen when an app is purchased. Instead, it downloads in the background, making the purchase of multiple apps much easier.
Apple still requires passwords for apps that cost money, and for in-app purchases -- though the 15-minute grace period allowing users to purchase multiple apps after entering the password once still applies.
Update: Apple appears to have reversed this change. When downloading a new free app, Apple now requires users enter their Apple ID password again before proceeding.