Moving is much more expensive during the summertime simply because it is the peak time for moving for the biggest majority of moving companies.
You can normally expect to receive a higher cost estimate during the summer because the movers may be booked to capacity, and therefore will charge more for moves.
To maximize your savings on your move as much as possible, try to schedule your move during the off season of moving, such as early spring, autumn, or winter.
You might be surprised at just how much waiting for peak season to be over might help you save.