* Cars with high-compression engines or turbochargers often *require* premium gasoline. These engines are designed to operate at higher pressures, and premium fuel's higher octane rating prevents pre-ignition (knocking), which can damage the engine. Using regular gas in these vehicles will likely result in reduced performance, potential engine damage, and possibly a check engine light. The owner's manual will explicitly state this requirement.
* Cars designed for regular gasoline will see *no benefit* from using premium. They are engineered to run optimally on regular fuel. Using premium will be more expensive and won't improve performance, fuel economy, or longevity. You'll simply be wasting money.
In short, the answer is: It depends on the car. Check your owner's manual. The manual will clearly state the recommended octane rating for your vehicle.