Typically, the driver would be "current limited" so that it would only draw the amount of current that it was set for. If your motor is rated for higher current, but you drove it with less than that amount, you would get less than the rated torque out of the motor.
The TB6560 boards with which I am familiar are rated for a maximum of 3.0 amps.
There are other motor parameters of interest, such as the winding inductance - lower winding inductance is better, because it means that you can achieve the optimum speed from the motor with a lower voltage. If you have a higher winding inductance, the voltage required to produce the optimum speed may be higher than your driver can handle, and so using a lower voltage means a slower max. speed.