Why is the combustion of steel wool similar to the corrosion of steel wool?

Both need oxygen and a main heat source like a Bunsen burner witch is a non luminous flame but you can use a wick candle and take a piece of steel wool and fluff it up a bit so oxygen can get in to the spaces to let the steel wool burn faster and more efficient because the oxygen feeds the flame. Then light the fluffed steel wool. You will see a combustion reaction with the steel wool, the oxygen and it's ignition source.

2Fe + 3O2 + heat = then make your observation

The same kind of reaction goes with the rusting of steel wool but it's just oxidization of the iron in the steel and to have a rusting reaction you will need a liquid compound such as water or some time gasoline will rust steel but the test you would do gasoline would not be a safe solution but water would work better because the natural sodium and other minerals would help in the decomposition reaction to the steel

So for the test take steel wool ( just about a half a gram) and jam it in the back of a long but narrow test tube the take a graduated cylinder and put about 50ml just enough for a precipitation reaction ( basically condensation) so the steel wool can have O2 and H2O to rust the steel wool. So put the narrow cylinder test tube face down in the water so the steel wool at the bottom of the test tube us faceing down and not in the water. So right now theirs a tight seal of O2 and H2O vapors allowing a decomposition reaction

Fe + O2 + H2O = observe the results may vary

The similarity is they both use a single decomposition reaction