If you are traveling on duty then check the special fare conditions when booking via HRG and you'll get your answer. The best 'perk', if you can call it that, is to use the diplomatic lane at US immigration with a valid NATO travel order.

If travelling for leisure, good luck, but it's never worked for me yet.

Edit to add: the new booking system does not allow the travel document to have a letter in it. If you are on a NATO travel order use only the digits from your service number or the computer will refuse to take it.