Weight is always a factor. knocking another 10 pounds off of a 2000 pound car is always a good thing. That's a half a percent improvement in itself.
But, that's not the main reason for a lightweight flywheel. There are a few good reasons, all performance related.
It all goes back to weight. The flywheel is "rotational mass". Anything in the driveline from the crankshaft to the gears in the transmission to the axles, brake rotors and the wheels is mass that has to be accelerated... and decelerated. Reducing the weight of any of those things reduces "rotational inertia". With less inertia, it takes less power to accelerate those rotating bits, and as Timinator said, it also takes less time for those bits to slow down when power is removed. It is generally accepted that reducing rotating mass is approximately equal to reducing 4x the amount of body weight in the car. So, if you can cut 10 pounds of flywheel weight, it's going to feel like taking 40 pounds off of the rest of the car.
So, when you're accelerating, some of your power (literally about 15% on the Mirage) goes to just accelerating all of the driveline components. Any reduction in rotational inertia reduces that load on the engine... it's like adding free power! Especially on lightweight and underpowered cars like ours, things like light wheels and a lightweight flywheel are common performance mods. While they don't "add power", they "free up" power. You make your wheels 25% lighter and your flywheel 50% lighter... you're reducing the rotational mass by quite a bit. You might reduce that 15% drivetrain loss to (drumroll....) maybe 12-13%. And you can actually SEE that on a chassis dyno. You're not "making" more power. If you had an engine dyno, it would show the same power. But, because you've reduced the drivetrain inertai, more of that power gets to the wheels, and you can totally see a 2% gain on the dyno from doing nothing more than changing the flywheel.
It's not night and day, but people pay big bucks to see an extra 10 hp on a 400 hp engine. "Dyno proven to gain 10 hp!" And it's only 2.5%. For us, 2.5% is a much smaller hp number, but it's still 2.5%! And every little bit helps. If I've gained 3% from a header, and another 3-4% from an exhaust, 1% from an intake... and another 2.5% from wheels and a flywheel... that means the car is 10-11% faster! All from simple little bolt-ons. (and yet... 10% faster than slow is still slow... dammit)
The other thing about a lighter flywheel, even if it didn't have the acceleration benefit, is that it makes it worlds easier to rev-match a downshift. You want to go from 3rd to 2nd? Clutch in, give the throttle the slightest nudge and the revs go right where you want them and you drop into 2nd. And because there's less flywheel mass to engage with the transmission, the clutch engagement is smoother. It's just generally more pleasant and fun to drive a car with a nice light flywheel. (couple that with a nice crisp clutch engagement, and... ooooooh!)
A lightweight flywheel is just another one of those things that's "greater than the sum of its parts". On paper, it doesn't seem like that big of a deal. But, in practice... especially on a lightweight car with minimal power... it's really nice.
But, there are downsides. Having less flywheel mass makes the car a little harder to launch from a stop. Most people adapt to this very quickly. You just have to give it a tiny bit more gas and/or slip the clutch a tiny bit more. But, it's certainly easier to bog a launch with a light flywheel. And some cars, particularly those with aftermarket superchargers, will do funky things at idle when the flywheel is too light. The computer tries to adjust the idle speed up a little bit (say you turned on the AC), it's going to overshoot because it's calibrated for the stock engine and it's got a supercharger on it. And then it overcompensates in the other direction and can set up an idle oscillation, or worse yet just stall. Stuff like that. It's not usually a problem on a normally aspirated engine, and usually not something you can't learn to live with even with a modded supercharged engine. (turn the idle speed up a little bit)
The other thing a lightweight flywheel will sometimes do, especially the aluminum ones, is "ring" at a certain rpm. The aluminum flywheel is sort of bell-shaped, and will resonate and ring at a certain rpm. Stock flywheel is a fat chunk of cast iron and doesn't do that.
For a gearhead looking for better acceleration or a more sporty feel, the positives far outweigh the negatives. If somebody made a $250-300 8-pound flywheel for the Mirage, I'd already have one!