In simple terms the power of a motor is in watts (or kilowatts).
Volts x Amps = Watts
Therefore to get the same power with a lower voltage you will need some more amps. That implies that a lower voltage battery pack will draw more current (amps) then a higher voltage pack.
The controller will deal with all that and deliver the maximum power it can.
Simply put, and this is very simply, the maximum voltage x the maximum current of the controller gives the maximum power it can deliver. You will need to consider controller design, efficiency, duty cycle, etc. but it is a sort of idea.
The difference between AC and DC is the way they work.
An AC motor rotates because the sinusoidal wave form of the current flowing into it creates a moving magnetic field that makes the armature turn round. Like a surfer on a wave, the faster the wave travels the faster the armature will turn. To increase the speed of the wave is to increase the frequency of the wave. The input voltage doesn't really matter too much except that the higher the voltage the lower the current.
A DC series wound motor is very different. The main drawback is that its speed is dependent on the input voltage and the load on the shaft. If you apply a voltage the motor will spin. The only thing stopping it from spinning faster and faster is the drag from the bearings and air resistance and any load you apply to the shaft. If you don't have a load then the motor can spin very fast in deed!
This is the reason why a DC series wound motor must NEVER be tested with more then 12v without a shaft load. I have a motor that will happily spin to 1800rpm on 12v. 24v will probably spin it to distruction.
When installed in a car the controller will limit the voltage to control the speed of the car. However, if you were driving along with your foot on the accelerator and decided to put your foot on the clutch, or select neutral, then your motor will spin to death.*
Anyway, the load on the motor will cause the motor to slow down and so to maintain the speed more voltage will need to be applied. The thing that causes the motor to reach and hold a speed is that the armature spining is causing a magnetic field to oppose the one that is driving the motor. This is the 'back EMF'. When the back EMF equals the input voltage the motor cannot spin any faster. To make it go faster you need to apply a higher voltage that will make the motor spin faster still until the back EMF balances it again. And so on. That is why you need high voltage to get speed in a DC car.
There is another way to get high speed in a DC car and that is to gear it for lower motor speed. This does work but you then need more torque from the motor which means more current.
You can look at a DC series motor like this: Voltage = speed, Current = torque.
The downside of this is that current also equates to heat and losses.
Running at higher current means the motor will get hotter. A hot motor is converting your battery power to heat instead of motion so the efficiency is reduced.
This is another area where AC and DC differ.
In a DC motor the heat is created in the armature, the spinning bit in the centre of the motor. That makes it difficult to keep cool and often forced air cooling is used.
In an AC motor the heat is generated in the static field coils that are around the inside of the motor casing. That means it can be cooled with big fins on the outside or a water jacket wraped around the motor.
AC is the 'ideal' situation reading back on this with the added benefit that it easily regenerates.
However, the downside is the cost of high power controllers. As mentioned above regarding the wave form that drives the motor around, it is actually three waves running 60deg apart in rotation, three phase. That means that the controller is really three controllers being controlled to make three waves exactly 60deg apart at anything from 0Hz (cycles per sec) to whatever maximum 1000'sHz is required. That makes it a difficult and expensive bit of kit to produce.
A DC controller is much, much simpler in comparison and so it is easier to get high power capacity and reliability in a cost effective package.
If anyone more knowledgeable can see errors in this, other then over simplification, then please flag them up and make corrections.
*Spinning a motor to death is a very bad thing. The centripetal forces on the armature are very high and when it spins too fast the copper coils and the commutator bars will fly outwards explosively sending shrapnal flying out of the motor explosively. On the DIY electric ar forum one chap was having his car towed home on a rope. The vibration on the road caused the gear stick to drop into first gear and the road wheels spun the motor so fast it exploded and buried bits of copper in the bonnet and in the road surface.