Amps x volts = watts
Flow x pressure = power
120 volts is plenty good for most household devices and accessories. To run a heating element large enough to heat a home or an A/C compressor, 240 volts handles the power much more efficiently. More voltage and less amperage runs cooler. More amperage generates more heat which is energy being dissapated into the air. A 120 volt motor running at the required amperage to match a 240 volt motor's output power, runs much less efficiently, is much larger, and will run much hotter and will not last as long as the 240 volt motor.
A tandem breaker, when overloaded, kicks both sides of the 240 volt hot wires. Each leg is 120 volts, if only one side had a problem in it's circuit, the tandem breaker will kick and shut power down to both sides. This protects the equipment from running underpowered or from damaging anything around it.
The powerlines out on your street are commonly 14,400 or 7,200 volts before it's reduced in the transformer in front of your home. Higher voltage is easier to deal with than higher amperage. If the power companies had to produce 120 volt at the required amperage, the wire would be as big around as you are tall. I think of voltage as pressure, amperage as flow, and watts are the resulting power.