On a car specifically, the flywheel rests at the end of the crankshaft on the outside of the crankcase. This is a source of stored energy, so that when you are taking off from a stop, there is instant torque applied once the clutch is engaged. The heavier the flywheel, the more angular momentum it will be allowed to store. And more torque it will be able to apply when the clutch is engaged.
The advantages of a light weight flywheel the movie are the same as reducing the weight of any part of your drive train, you are going to get a quicker revving engine. This is fantastic once you are moving, because it means you will get to the higher RPMs, where the power is, much quicker. The down side of this is drivability.
A lightweight flywheel means you are going to have to get really good at clutch control really fast. The stock, heavier flywheel means you can be pretty liberal with clutch engagement and still not stall. However, the lighter flywheel means as soon as the clutch engages, the clutch, driveshaft, rear end, axles, etc that are at a dead stop can overcome the lower angular momentum of the light flywheel, thus causing a stall. The solution, of course, is easy, you add more throttle while engaging the clutch. This can, however, cause added wear to your clutch and replacing clutches gets old after a bit and can be costly.
The other downside of the lightened flywheel flexplate comes between shifts. Once the clutch is disengaged it means the engine is going to rotate slower, because the heavy flywheel is not there to keep the momentum going. This means a greater RPM drop between shifts if you shift at the same speed that you did with the heavier flywheel. This could mean your engine falls out of the power band between shifts.
If your car is just a daily driver, there is no need for you to install light flywheel. There are probably better, easier, and less expensive ways to make up the performance difference without the added headache of driving with a light flywheel and clutch. However, if your car is a racer only, then yes, it’s kind of a no brainer.