Why long-distance movers quote by weight
An interstate move is priced from the weight of the shipment and the distance travelled, which is why a quote asks how many rooms you have and then estimates a weight. The estimate is a rate applied to a guess, and the guess is checked on a weigh station scale on the way.
Anything that adds weight adds cost: books, tools, a piano, a garage full of things nobody has opened in five years. The cheapest way to reduce a moving bill is to own less by the day of the move, and the second cheapest is to pack it yourself.