From a physics POV:
The screwdriver is sonic. That means it works by emitting sounds of the resonant frequencies of objects and vibrating them in such a way that work gets done. It seem a bit advanced in that it can affect electrical systems too, and it seems to have the capability of doing the exact calculation of the sound required on its own. It can scan by using a system similar to sonar, which will help it pick up resonant frequencies in the first place.
Metal is easy, a single metal part has a couple of well defined resonant frequencies and can be targeted easily.
Wood is not. Wood is organic, and as such a wooden part is much more complex when it comes to finding resonant frequencies (or even having resonant frequencies). There's also extra damping, so any resonant frequencies you have might not be effective.
The screwdriver does work on Flesh, though, which is organic. This might be due to the Flesh internally crystallizing to form the solid body (liquid crystals and similar substances are easier to resonate). Alternatively, it might have some nanobot-like components.