do the voltage and amperage applied to a motor matter as long as the wattage stays at what its rated for?
Yes, the voltage and amperage do make a difference. 

To start with the motor wants to spin at a speed proportional to voltage, so if the voltage is too high or too low the motor will either over-speed or not run fast enough. Over-speeding can produce excessive brush arcing and commutator wear, reduce bearing life, or even cause the armature to fly apart. 

Secondly, 'copper' loss (caused by resistance of the winding wires) is proportional to current squared, so running higher current at lower voltage results in higher loss at the same input power. This is only partially compensated by the lower 'iron' loss (caused by hysteresis and eddy currents induced into the magnetic core) at lower rpm. If the motor has a built-in cooling fan it may also run hotter due to less airflow at the lower speed. 

If the battery voltage is higher than the rated motor voltage then you can reduce top rpm by limiting the throttle to less than 100%, but the speed controller must still be able to handle the full motor current (which with a PWM controller will be higher than the battery current).

Bottom line is you should find out what voltage the motor is designed to run on. One way might to run it on eg. 12V and measure the rpm, then relate that to what rpm it ran at in the go cart.