On a modern NA engine it will depend on the "computer". If the engine advances timing to take advantage of the higher octane there will be an improvement. If the computer programming is based on lowest common spec (lowest spec octane) then no. This last one is likely the case on a NA car spec'ed for low octane fuel but not always.
Over simplified above but the computer senses pre-detonation/knock/ping (not actually octane) and adjusts (retards) timing to get rid of it. Forced induction engines do the same but have boost and timing in the adjustment bag, still won't help unless it is designed to increase these to take advantage of the higher octane fuel. Other things can also impact this (engine temp, air temp, etc) but the tool is to retard/advance timing (or boost) based on sensor outputs. Getting more power will depend on the ability to advance these things beyond the base fuel spec.
In contrast a NA performance engine may be tuned based on the high spec fuel the manual calls for but if you put 87 in it will sense the pinging and retard timing to not ventilate the pistons. In the pre-computer era you could manually rotate the distributor to increase/decrease timing and set the engine up for the fuel you were running (more timing advance). Without fancy equipment some trial and error with your ears and seat of your pants resulted in it dialed in.... and a handful of more power.
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