While wheel/tire weight is a big issue, SCC, Grassroots Motorsports, et. al. miss the reason that older vehicles generally handle worse with lower profile tires - shocks.
A tire is just another spring in the suspension with a rate of 1,000 to 1,500 lb. in. depending (mostly) on inflation pressures. That sounds like a huge rate but tires respond much quicker then the suspension can. If you elimiate some of that response via higher air pressures (required for shorter sidewalls) you have to get back some of that response in the only other place possible, bump and rebound damping. If you don't get that response back, you get a car that tends to skate. (BTW... that's why your car will always stick better with the lowest air pressures you can run without "rolling over.")
Modern performance vehicle shocks are designed with more compliance in bump and control in rebound. The shocks can respond better and provide the compliance necessary to make ultra low profiles work.
Unfortunately for us older car owners, we're stuck with 1970's shock technology even when we purchase Tokico Illuminas, Bilstiens, etc. designed for our cars. The shock designers rely on the tires to provide the responce and compliance and build their shocks with too much bump and not enough rebound.
Now, if you had, say, a set of Penske 8760 triple adjustbale shocks on your 240Z...