Lower octane fuels are just more volatile, they aren't really dirtier, they actually burn faster. Higher octane burns slower & can withstand more pressure. So you can actually end up with a more complete burn using lower octane, depending on the engine.

Higher octane is better for higher heat & higher compression. Lower octane CAN be better for lower compression & large combustion chambers (like rotary engines). But there are a lot of variables. Cooling is a big factor too.

If you put 110 race gas in your average road going consumer car, you are just wasting money. The tune will be too rich & it will likely be worse for the car. But you cannot run a 13:1 compression race engine on 87 or 89 octane, it would likely blow up from pre-ignition or detonation.

Modern cars will generally advance or retard the timing & even adjust fuel maps based on the fuel you choose, so as long as you choose between the normal 87-93 or maybe 95 octane then the car should adjust & the power & fuel mileage may change a little, but it won't be significant. If you have a high compression honda or something turbocharged it may make a bigger difference because the computer may have to actually retard things to protect the engine from low octane gas.