The app is awesome - especially if you have kids who order things custom. You just put it in yourself, pay securely, and give them a code. Literally all they do is ask you if you want a receipt and hand you your order, it's awesome.
In addition to the deals that were already mentioned, you also accrue points and get free crap with it. Our youngest loooooves McDonald's, and we ended up getting it after baseball games with undesirable amounts of regularity, and it was great to just start the order at the ball park and have it ready when we got there.
On the original topic - I'd much rather do the QR codes, assuming they have a mobile site that sizes accurately. No sticky menus covered in snot, etc., they can update as needed if out of an item, can add item of the day. It is way more convenient and environmentally friendly. Would I want it in a nice restaurant? Probably not - but kids aren't going to be snotting all over a paper menu at Django or something usually. At a burger place, 100% QR codes, please.