No, your Gap insurance will not pay the difference between what the loan payoff on your vehicle and the trade-in value you would get if you traded it in on a new car. Gap insurance is there to cover the difference between the actual cash value (ACV) of your vehicle and the loan balance when your car has been totaled in a covered loss situation.
If a loss occurs, GAP insurance will pay the difference between the actual cash value of the vehicle and the current outstanding balance on your loan or lease. Gap Insurance protects your vehicle lease or loan in this way, not if you are trading in your vehicle and still owe on it.
If your vehicle has been totaled by accident, theft, fire, flood, tornado, vandalism, or hurricanes your primary insurance company typically pays the actual cash value. It is often considerably less than the actual amount you still owe on your loan or the amount due for a lease payoff. When you have negative equity in a car (are upside and owe more than the worth of the car) and your car has been totaled out is when Gap insurance comes in handy.
Also keep in mind that if you roll in the amount you still owe on your current vehicle into a new car loan and get Gap insurance on that vehicle that this old loan amount will not be paid out by a new gap policy. Gap insurance excludes payment for carry-over balances from previous loans or leases.