Simply put it is down to lightness, low engine output, lack of electronics and unforgettable design.
Sports cars and hot hatches from the 90’s typically weigh under 1000kg (Peugeot 205 1.9 GTI weighs approximately 875kg), compared to a kerb weight of over 1500kg for modern equivalents. The weight increase of 50% has meant that modern cars need much more power to perform. The inertia effects of all the extra weight in corners has a huge impact on handling. Grip has had to be increased to compensate and also electronic aids such as traction control are needed to assist the driver keep the car on the road (and pointing in the right direction). The simplicity of having a light, low output car negates the need for all this assistance. Gordon Murray (Maclaren F1 designer) is a champion of reducing kerb weight and engine outputs. His supercar weighed around 1100kg when most rivals weighed significantly more.
The size of cars has ballooned, an excellent example of this is the change in size of the Nissan Skyline from the R32 version up to the current day R35 GTR. Engine output has increased from 2.6 litres in the R32 ’89 up to 3.8 litres in the R35 ’07.
I believe electronic aids and assistance have dulled the driving experience and skill required by the driver. Cars are faster now but also in most cases, the driver has less impact on making the car drive fast. The extreme of this is the Nissan GTR (Skyline), which in it’s current R35 guise has electronic aids that control every aspect of the handling and dynamics. Whilst most sports cars are now extremely accessible “turn the key and drive” machines, this takes away a level of responsibility and learning which is essential when driving an older equivalent. Learning how to drive an older sports car fast creates a bond between the car and driver that cannot be matched by modern sports cars.
The birth of the mass produced modern electronic aided sports car occurred in the mid 90’s. There was a performance revolution, extreme saloon cars such as the Mitsubishi Lancer Evo and Subaru Impreza really gave high performance sports cars something to worry about. Both could comfortably outpace a 911 Carrera around a track with less reliance on driver skill.