The preference as between wood and wooden varies in English writing depending on the noun being modified. This is perhaps most easily illustrated by looking at a series of Google Ngram charts that match different adjective-noun combinations. In the case of modifiers for the noun door, as the Ngram graph that fev's answer cites, the preference for "wooden door" (red line) over "wood door" (blue line) was remarkably stable for almost 200 years, but has shot up impressively since about 1980:

The plots for "wood floor" (green line) and "wooden floor" (yellow line), overlaid on fev's original Ngram chart, are somewhat less consistent and indicate a somewhat weaker preference for wooden over wood:

Even more inconsistent are the results for "wood frame" (navy blue line) versus "wooden frame" (purple line):

In this case, "wood frame" was actually more frequent in published writing for a couple of decades (roughly 1970–1990), although "wooden frame" has more recently reasserted itself and the incidence of "wood frame" has dropped off considerably. (I should acknowledge that "wood/wooden frame" often appears as a compound modifier for a following noun such as house, which sets it apart from "wood/wooden door" and "wood/wooden floor"; however, that difference in application doesn't seem to me to be critical to the choice between wood and wooden.)
A couple of things emerge in this series of Ngram charts. First, the frequency of the "wooden" forms has increased sharply in the case of door and floor and significantly in the case of frame over the past 40 years, while the frequency of the "wood" forms has increased impressively in the case of floor, grown modestly in the case of door, and dropped by more than half in the case of frame during the same period. This suggests that we have entered an era of increased preference for wooden over wood, at least with regard to these three pairs of terms.
Second, the current degree of preference for the wooden term over the wood one seems to depend to a great extent on the following noun. Of the six terms tracked in the third Ngram chart above, "wooden door" is the most frequent and "wood door" is the least frequent, indicating a very strong preference for the former. Both "wooden floor" and "wood floor" have enjoyed upswings in popularity in the past three decades, although the plot for "wooden floor" has risen more sharply. And the frequencies of "wooden frame" and "wood frame" have gone in opposite directions since 1980, from a state of rough equality to one in which "wooden frame" is more than twice as frequent (although "wood frame" is still more frequent than "wood door"). The Ngram plots for "wood fence" versus "wooden fence," "wood table" versus "wooden table," etc., show similar profiles, with the "wooden" form more frequent than the "wood" form, and with the frequency of the "wooden" form increasing markedly in recent decades.
I don't know why the movement toward "wooden" forms is occurring. It is true that wooden always either literally or figuratively means "made of wood" whereas wood sometimes means "made of wood" and sometimes means "made to act on wood" (as in the case of "wood saw" and "wood glue")—but there are few instances where any serious ambiguity might arise from using "wood [noun]." The only one I can think of is "wood carrier," which might mean a carrier made of wood or a carrier for transporting wood.
The areas where "wood [noun]" dominates "[wooden] noun" are those where the sense of the term is "[noun] composed of pieces of wood"—for example "wood pile," "wood fire," and "wood shavings." In none of those specific instances does the "wooden" form sound right to me. Sure enough, we get this set of Ngram plots for "wood pile" (blue line) versus "wooden pile" (red line), "wood fire" (green line) versus "wooden fire" (yellow line), and "wood shavings" (light blue line) versus "wooden shavings" (maroon line):

The upsurge in instances of "wooden pile" in the late nineteenth and early twentieth centuries appears to involve occurrences of "pile" in the sense of "slender column or pillar driven into the ground and used to bear a vertical load" rather than in the sense of "stack or rick [of logs]." The blips in frequency of "wooden fire" during roughly the same period seem related to instances of such longer phrases as "wooden fire-box," "wooden fire escape," and "wooden fire doors."