A lightweight flywheel gives you more torque at the wheels. Because for the same combustion, less of that energy is absorbed by the heavy flywheel, and gets to go directly to the wheels via the crank and gearbox etc.
More torque at a constant RPM means more power at that same RPM. More torque at an engine's highest RPM, means higher power. A lightweight flywheel does increase the engine's top power output.
Trev also proved this at the Melbourne dyno runs. He beat all the other NA micras that had the same mods, he was the only one with a lightweight flywheel with those same mods (extractors, intake etc).
If a lightweight flywheel on a given car somehow dictated that the torque at the peak RPM was the same, well then yes the peak power would remain the same. But this shouldn't ever be the case because whatever energy the motor output before, its now going to have more at any RPM because we've just removed a heap of rotating weight out of the system, allowing more of the engine's effort to go directly to the wheels.
So I'm backing the claim that lightweight flywheels do increase both torque and power. Power = torque x revs.
As for fuel economy, I agree with White Knight and aj somewhat, that its more complex than a simple function. I think it depends on a whole heap of variables, driving style, roads, conditions, car, engine capacity and percentage of power/torque increase that the flywheel brought compared to what power/torque the car had previously. Yeah less effort is required by the engine to make the same power, so for accelerating and getting up it, I would have thought you would save fuel as aj says, but then fuel consumption is also a function of power being made. So if you are making more peak power, depending on a car's ECU, its *possible* that it could feed in more fuel - maybe for safety margins or whatever its reason may be.
But when you're cruising, you're having to keep the foot further down so that car can hold the same momentum, so are you infact using more petrol just to sustain the same constant speed while cruising.. Or depending on gear selection, you might be able to hold the same speed in 5th gear now that you used to have to stay in 4th gear for. Which would indicate that you will be saving fuel just because you can stay in 5th gear instead of 4th.
Interesting!