Most new cars are now either high-compression naturally aspirated, or turbocharged. Both of which benefit from increased octane.
If you have an old, or economy car, with lower compression, you will in fact see a bit more horsepower and lower emissions from lower octane, as it can burn more completely in an inefficient motor.
Newer cars with knock sensors will adjust the timing to be able to run better with better gas, many even have sensors now (flexfuel, etc).
The result is usually better power and mileage for turbocharged engines, and slightly better mileage (maybe not enough to offset price) in a natually aspirated one, and slightly better emissions.
Be aware USA and Europe measure octane differently.
Tldr - Buy what your gas cap says.