I had quite some time to experience the difference, having run lightweight for some time, rubber on carpet..Protoform Speed 6 and LTC-R
I found the lightweight shell made the car rotate far too much, too early in the corner, making it really unstable and difficult to set up. Too reactive for my (slightly edgy) setup it just ruined the balance of the car.. By contrast the regular weight made for consistent, predictable handling with plenty steering, which was smoother, and much faster. Just right.
Perhaps the results are much better for asphalt.
Well that's my experience, perhaps others have something more positive to say.