The capacitor does not soften the motor start up. Put simply, on a single pole motor the capacitor causes one of the poles to lag slightly when the motor is first started. This determines the direction that the motor will rotate. Without it the direction the motor turns would be random. Once the motor has started turning, a centrifugal switch will remove the capacitor from the motor winding circuit. Some smaller sinlge pole motors will have a shaded pole instead of a capacitor - this is usually a copper bar that shorts a portion of the laminated core. This is called, naturally, a shaded pole motor. Shaded pole motors are cheaper to manufacture, but are less efficient and weaker than a capcitor start motor.
Sometimes the capacitors just go bad, but I recommend that you check the operation of the centrifugal switch.