Airplane Mode basically turns off All radio transmissions, like ckuan said, wifi, cellular and Bluetooth are turned off. Thus no Internet, no listening to music via Airplay or Bluetooth, only internal speaker or plug-in headphones will work.
However, you can turn on wifi and/or Bluetooth and still be in airplane mode. This means that cellular is still turned off.
Do Not Disturb is a whole different cat. Say you don't want to be disturbed at night time when you sleep. You'll turn DND on and everything coming in, calls, text, weather alerts and any other notifications won't sound/vibrate or turn on the screen. But if you have an Alarm set in the Clock app, that will continue to "wake" you in the morning. As a matter of fact, that will by pass the mute switch as well. Also, you can change the default settings of DND in Settings---> Do Not Disturb---> Allow Calls From---> Everyone, No One, Favorites or a particular Group of Contacts. Also, if a caller called you twice with in 3 minutes. The DND will be bypassed. Even if they are not on your list from above. Repeated Calls has to be turned on for that as well.
KOT