To me, the difference is subtle but real.

If you see something on the menu like $10 off on Thursdays, or BOGO, that becomes the price of the item. (Your $29 entree is now a $19 dollar entree), or in the case of a BOGO one will be listed on the bill under some variation of $0.

When you order a meal and present a coupon, the coupon becomes your form of payment, not a discount on the price. In the example above, if you had a $10 coupon, the price of the Entree is still $29. You PAY the total bill using your $10 coupon as part of the payment.