A higher octane rating results in smoother engine running, it is a measure of how prone a fuel is to pre ignition, the higher the rating the better. In terms of performance, as already mentioned, a poor octane number results in pinking, or knocking, this throws off engine timing, results in poor compression, and causes excessive wear and damage to an engine even over relatively short periods.
The better the octane rating of a fuel, the better the compression ratio that can be used, an important factor when determining engine output and directly affecting relative engine performance.
Piston engine performance relies on more than a simple chemical interaction of hydrocarbons and air at the cylinder face. For the technology available during the early part of WW 2 the performance advantage of the higher octane fuel was significant. It allowed better compression ratios, meant lower maintenance over heads due to reduced wear and allowed engines to perform as close to optimal as was possible, for longer periods of time.
While water injection helps offset pre ignition, it also means the engine is far more susceptible to damage, and brings an entirely new set of problems to the table when considering fighter aircraft.
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