This is a classic question, to be found in almost every Ask-a-Scientist collection. There will not be a difference in weight. While flying, the birds' wings exert a downward force on the air in the truck exactly equal to the weight of the birds, and the air will transmit this force to the floor of the truck, and hence to the weighing mechanism.
The only way to beat the weigh scale would be to have very highly trained birds, which would fly to the top of the truck and, immediately before the weight is recorded, fold their wings and fall. If the weigh scale is read in the short interval before the free-falling birds strike the floor of the truck, it will register only the weight of the truck.
(A similar question involves a 100-kg juggler, three 5-kg juggling balls, and a bridge that will support exactly 110 kgs.)