Do not disturb turns off notifications while the screen is off (except for any exceptions you grant, like two calls in 3 minutes, or calls from favorites or certain contact groups). In DND, the phone doesn't turn on the screen or even buzz.
In silent mode, the notification comes through, the phone vibrates (if vibrate on silent is enabled), and the screen turns on to display the message, but it does not play the sound (unless it's an alarm set in the clock app).
I'm definitely loving DND. I have it set to allow calls and texts from certain family members and that's all. So if one of the kids needs me in the middle of the night I get the call or text (custom message alert) like normal. Everybody else can wait. And the automatic feature means I just plug my iPhone in next to my bed and never have to set it. Once you have it set it goes into effect at the same time every night.