There are two possible effects from my point of view: the wood doesn’t heat the same way metal does, so it may have an impact on this ; and the wood is porous, so the water and stuff will not interact with the surface in the same way it would do with metal. But I doubt both those effects are that important.
Wooden spoons tend to be bulkier so maybe that’s why it’s always said to use a wooden one.
There are two possible effects from my point of view: the wood doesn’t heat the same way metal does, so it may have an impact on this ; and the wood is porous, so the water and stuff will not interact with the surface in the same way it would do with metal. But I doubt both those effects are that important.
As you say, maybe the thickness matters.