Batteries approximate to a constant voltage source, a capacitor doesn't. For a capacitor..
Q=CV
I = CdV/dt
so
dV/dt = I/C
So the voltage "curve" depends on the load current. If you discharge at a constant current the voltage falls in a straight line. If it's a constant resistance load you get the classic exponential curve.
If the load must get a constant voltage (not all need that) then some sort of voltage regulator, perhaps a switching type, can be used to provide it.
You don't always need a voltage regulator. The motor in an electric car for example doesn't necessarily need a constant voltage but it does need some form of speed control. It may well be possible to build one control unit that accepts the variable voltage from a capacitor and outputs whatever the motor requires.