What is the difference?
Does the oil protect the nuts from oxidation?
Does it make it taste better?
Does the type of oil / nuts matter?
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The primary reason to toast nuts with oil is to get better heat transfer from the pan/oven to the nuts themselves; this is particularly important if toasting nuts on the stove, as you can end up with very dark brown spots on otherwise raw nuts, where a small part of the surface was touching the pan. Oil helps to increase the effective area of heat transfer, jut like it does in any sort of frying. You can toast nuts without the oil, but you have to be extra vigilant about keeping the nuts moving so no one spot gets too much contact with the pan.
Of course, adding oil also adds the flavours of the oil - usually one would use a neutral oil to toast nuts in, but if you were looking for a specific effect there'd not much stopping you from toasting in something distinctive.
In terms of protecting the nuts from oxidation, a coat of oil will technically block oxygen from getting to the nuts themselves, but all that achieves is that instead of the oil in the nuts going rancid from oxidation, the oil in the coat of oil goes rancid first, which isn't any better from an 'I'd like to eat this' point of view.
There is no cooking advantage to oiling nuts while roasting (that is, cooking them with dry heat in an oven). Adding oil to the nuts during roasting is done strictly for flavor, and is only appropriate for nuts intended to be eaten straight or as a garnish, per The Kitchn:
But it’s not always appropriate to roast nuts with oil, especially when they are being used in a baking as the oil does increase their oiliness and can throw off the recipe. I roast nuts in oil when I am adding them to a salad, for instance, when they will be used as a garnish, or when I want to serve them with an aperitif.