Intuitively, wooden board makes more sense, for wooden is an adjective meaning "made of wood," whereas wood is both a noun and an adjecitve. Google Ngram viewer also seems to suggest wooden board is the more common term:
However, I have seen wood board many many times. For instance: wood boards/wood slabs, wood frame, and wood board AND wooden board. What are the differences?
By some formal grammars, wooden is the adjective, and wood is the noun. Nowadays, any good descriptive reference will note, as you have, that wood can also be an adjective. Similar patterns exist for other materials - golden and gold being the one that comes most easily to mind. The same pattern of the noun now being recognised as an adjective as well applies in all such cases, as far as I'm aware.
Where there is some phrase that has been in continuous usage for a long time, we tend to keep using what we have always used. So while a wooden figurine might also be called a wood figurine (though the adjective wooden has more usage there than the adjective wood, according to Google ngram), a board made of wood is still a wooden board almost invariably.
However, wood board still shows usage, and has done for a long time. That's because it's a potentially valid construction even if wood is always a noun. There's a way of using nouns almost like adjectives, one that learners should be careful about using because it has many pitfalls. It's called using a noun attributively. An attributive noun looks like an adjective, but it isn't. It's a noun used to indicate an attribute of another noun, it is placed before the other noun and generally has a meaning equivalent to that achieved by putting the word after the other noun and putting a preposition between them.
A physics degree.
A degree in physics.A chess game.
A games of chess.A pancake recipe.
A recipe for pancakes.
Thus, a "wood board" is a "board (made) of wood".
