The wood type in any stringed instrument matters a great deal, especially on acoustic instruments. Some parts of the violin contribute more to the overall tone quality than others, but all the parts make a difference.
A stringed instrument is a case study in engineering trade-offs. After all, how does a violin produce its sound? To begin with, note that the body of a violin is basically an empty box full of air. The sound you hear a violin producing is the result of that air vibrating and escaping out the F-holes of the instrument. How does this happen? Drawing the bow across a string causes that string to vibrate. The vibration of the string causes the bridge to vibrate, which in turn causes the top of the violin to vibrate. The vibration of the top of the violin is passed through the sound post to the back of the violin, and the two parts---the top and the back---now both vibrate. This causes the air trapped between them to vibrate and get pumped out of the F-holes into the surrounding air, and that vibration is what you hear.
So, you want the wood of the violin to transfer the vibrations of the strings to the box of air as efficiently as possible in order to produce the most sound, and so you want the wood of the instrument to be light and resonant. Maybe a nice light balsa would do the trick?
But no luck, because of the inherent engineering challenge: While you want the violin to be as light and resonant as possible, it also has to withstand the significant string tension of four strings pulling on it and trying to fold it up. If the wood is too light and weak, the violin will collapse like a cardboard box, which obviously is no good. If you build the violin to be too heavy and sturdy, though, it will absorb and reflect the strings' vibrations rather than passing them along to the air. It's a tough problem to solve.
So to build a violin, you want wood that has all these qualities: it has to be light, resonant, strong, stiff, and sturdy. These are difficult qualities to find all in one species of wood, but a few kinds have proven to be effective: cedar, spruce, mahogany, walnut, maple, etc. All of these woods work well for acoustic stringed instruments, but they each behave slightly differently when transferring the strings' vibrations into the body of the instrument.
The point is: The entire acoustic mechanics of a violin (or any stringed instrument) completely depend on the ways in which the various pieces of wood transfer the vibrational energy of the strings into the box of air. Different kinds of wood resonate differently and thus transfer that energy differently, and as such, the kind and quality of the wood of a violin makes all the difference in its sound.