Weight is a significant factor in transportation energy storage because the more something weighs, the more difficult it is to accelerate (after a stop), and hence the more of the fuel gets used initially for transportation of the remaining fuel (used later in the trip). This is less important in some applications (e.g. freight trains, where the weight of the cargo hugely overwhelms the weight of the fuel), but for most transportation it is highly significant in determining a vehicle's maximum possible fuel efficiency. It is especially significant in aviation, where any increase in mass leads to a significant increase in air drag.
Volume is also important, however, in that the storage space for the fuel should not take up more than a small fraction of the vehicle's available cargo space; a fuel with high energy-per-mass but low mass-per-volume will require a large tank for storage, which is inconvenient and can also weigh a lot and counteract the benefits of the fuel's energy-per-mass.