Motors have the advantage that they can produce power at a wide range of RPMs while still retaining their efficiency.
For instance, a reciprocating engine cannot run at 0 RPM, but a motor produces its most torque at 0 RPM. Also, by varying the current going through the motor, it is easy to control the car's acceleration without shifting gears.
If you take an automobile Gasoline engine as an example and write down what essentially it does:-
Basically a Gas engine produces a lot of RPM but a rather small amount of torque, hence if you connected your flywheel directly to your driveshaft your engine would stall immediately. It would do this for two reasons:-
The Otto cycle is a reciprocating cycle, i.e the engine cannot do anything from stopped. When the vehicle is stationary it must be disconnected from the transmission to prevent the vehicle stalling
A gasoline engine produces little useful power or torque at idle and has a definate "power-band" where the engine is at it's most productive. This is usually between 2000-5000RPM
If you take RPM at the wheels for a given speed on say a car with 14 inch diameter tyres.
First I'm converting to metric so 14 x 2.54 = 35.56cm
Pi * 35.6 = Circumference of the wheel (distance covered with each revolution)
3.1517 * 35.6 = 112.2cm (or 1.122m of distance covered in each revolution)
Hence for a speed of 60km/h (roughly 40mph) (or 1km/min)
Distance covered is 1000 metres per minute, therefore 1000/1.122 = 891.265 RPM
As you can see the required RPM at the wheels is less than most engines idle at, doubling the speed to 120km/h (roughly 80mph, probably closer to 75mph guessing off the top of my head) would only require a wheelspeed of 1782.5 RPM.
Therfore we have to design transmissions with the limitations of Otto cycle engines in mind. As torque and RPM are interchangeable through gear ratios this isn't much of a problem.
Electric motors however are a completely different ball game, you could easily be able to buy an electric motor that would be able to give you enough torque throughout the rev range from 0RPM to 2000RPM (If the wheels are 14inch remember that equates to 0-80mph) without having to resort to gearing. The lack of a transmission would also improve power efficiency as power wouln't be lost in the transmission.
The biggest problem with electric powered cars has been the power source, taking a 150 horsepower motor is equivalent to around 110 Kilowatts.
Power = Voltage * Current
110,000 watts = (x voltage) * (x current)
from the amount of power needed, the numbers involved in creating that power from volatge and current need to be large also:-
110,000 watts = (110 volts) * (1000 amps)
110,000 watts = (11,000 volts) * (10 amps)
or whichever you like it, you can see though that providing a portable electrical power source that would power a 150 horsepower motor has proven to be incredibly difficult. And to think, you get that much power from a tank of Gas.