Higher octane gas is ment for higher compression and forced inductions engines. Higher octane gas doesn't always burn hotter, but burns at a higher temp (it takes more heat for it to start to burn). The simplest way of explaining is that high compression and forced induction engines generate more heat. Both systems need the higher octane to prevent detonation (when the gas ignites before the piston reaches the top of it's stroke). Detonation will cause an engine to have a pinging sound and/or blow up. It also drastcily drops the engine's performance. This is because when the piston is still moving up, the gas has already ignited and is trying to push down on the piston. (causing the engine to ping, blow up, and/or loose performance)
You can run higher octane in a vehicle that doesn't require it, but not visa versa. However, all the higher octane will do, in an engine that doesn't require it, is make your wallet lighter (as previously mentioned). In therory, if the octane where high enough with not enough compression, heat, or spark, the fuel would not ignite.
I know there are other things to consider, like timing, the rate of combustion, combustion tempurature, etc, but this is the simplest explanation.
