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A lighter flywheel will give you economy on stop and go. It is basically the same principle as lighter tires.
One disadvantage is that your engine will be easier to stall, and to not make it stall you would have to give it more gas and your fuel consumption will increase a bit untill you get the hang of it (and maybe even after that as you still have to rev a bit higher).
Once evertyhing is up to speed it will not make a difference.
I think (i am not sure about this) but you do need some kind of rotating mass (for energy storage) so that the engine has enough inertia to compress the cylinders. The mass of the crankshaft or the pistons may or may not be sufficient for this.
Disclaimer: i have never built/rebuilt an engine, this is just based on what i have read on the internet. But i am an engineer.
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