If you have a 24V PSU, then just using a single ended driver will give you +24V on one end of the motor and either 0V or 24 V on the other so a total voltage drive of 24V. If you use a H bridge, at one instance the motor gets a 0V on terminal A and +24V on terminal B, then +24 V on terminal A and 0V on terminal B, i.e. the motor will have 48 V of drive on it.
With no cross over the motor will not run, because there will be a current path to the 0V line but both the transistors going to the +24V line will be switched off.
Frank