If a car has a "knock" or a "ping" when it runs, filling it up with higher-priced premium gasoline will probably eliminate the knock or ping. Otherwise, the only benefit of premium gas is that the detergent and other additives in the higher-priced premium blend might help keep the carburetor and fuel system clean.
A car's combustion engine has pistons that fill up with a mixture of gasoline and air, driving them up the cylinder. When a piston reaches the top of the cylinder, a spark plug ignites the gas-air mixture, which sends the piston back to the bottom. When the gas-air vapor ignites without the spark plug, it results in engine knock, which is very bad for the engine's health. High-performance engines, engines in older vehicles and very heavy automobiles require a high compression ratio and might knock if premium gasoline is not used. Most other cars are designed to be run on lower octane gasoline, and there's really no difference whether regular unleaded or premium unleaded is used.
Every car might potentially benefit from the additives that are put in many premium gasolines. The additives include detergents and solvents that help keep the carburetor and the fuel system clean. It's unclear, however, whether the additives are worth the price difference between regular unleaded and premium unleaded