I don't know what's the difference? when would you use do not disturb?
Silent is when you don't want any noise from your phone. Now, Apple sees fit to exclude alarms from silent, which I don't care for because you can have your phone on silent and still have an alarm go off at the worst possible time. But in any case, that's what silent is for.
Do Not Disturb means any notifications are silenced. Calls, messages, email, anything that is a notifcation.
DND allows you to specify certain people though that can effectively ignore DND and therefore their calls, messages, etc are not silenced.
Admittedly, there really isn't much of a difference, but don't tell Apple that.
Now if someone could make something that kills alarms too I'd be really happy. Silent is supposed to make the entire phone silent with no noise. Right now if you truly want your phone silent there are several steps. Turning DND on, turning on the mute switch and then disabling alarms.
There's two camps on that though, which is why alarms are never silenced.
Compare this to my 2009 HTC Touch Pro that was entirely silent with one push of a button. Yay Apple!
Not exactly right on several points.
Silent: The phone will not make noise (apart from alarms), but if you have it set to Vibrate while silent, your phone will still vibrate and your screen will light up with any call, text, or other notification.
Do Not Disturb: Your phone will not light up, make a sound, or vibrate for any notification. The only exception is if you allow Calls from your favorites to come through while Do Not Disturb is on (only calls will come through, Texts/messages will not). That, or if you allow the situation where someone calls twice in several minutes, the second call will come through.
If you just turn your phone onto silent, say, at night, it will still vibrate and the screen will light up. Turning on Do Not Disturb eliminates those annoyances.
If you want your phone to be truly silent, you just need Do Not Disturb (and to turn off alarms if necessary). There's no added benefit from turning on the Silent switch.
Also, I'd say the biggest use of Do Not Disturb is to not be disturbed at night while sleeping..having an alarm set to wake you up is still crucial, and a very logical "disturbance" to be allowed. If you have so many alarms set that it's an issue, well then..