The usual story is that his first use of a wooden sword in a serious duel was when he defeated Sasaki Kojiro. Sasaki Kojiro was famous for using an extra-long sword, and on his way to the duel, Musashi carved an extra-long wooden sword from an oar, longer than Kojiro's sword. With this extra reach, Musashi was able to kill Kojiro with a blow to his head, while Kojiro only managed to cut Musashi's head-band, the blade not reaching Musashi himself.

This illustrates one potential advantage of a wooden sword: it can be longer than a steel sword, while remaining lighter, providing an advantage in both reach and speed. These advantages are the foundation of jo (a 4' staff) against sword; the same advantages result from an extra-long wooden sword (but without being as adaptable to spear technique as a jo - a lot of jo technique draws on a combination of sword and spear technique).

A wooden sword (or a jo) is less lethal than a sword - a light hit with a sword can cause a lot of damage, while a light hit with a wooden sword, or even a moderate hit, might do little beyond minor bruising. Which brings us to a particular point about Musashi: supposedly his first victory in single combat was at the age of 13 when he beat an armed samurai to death with a stick. Basically, Musashi was reputed to be immensely strong, and was already capable of killing with a stick at 13, and only grew in strength as he matured. Musashi's great strength made him more lethal than usual with a wooden sword.

Of course, it's hard to be certain about the truth of these details, but they're part of the usual story about Musashi. What is certain is that a long light wooden weapon does have an advantage in reach, and can have an advantage in speed.

The Victor Harris translation of Musashi's Book of Five Rings gives the common versions of these events. (The Kenji Tokitsu translation gives a more detailed chronology, but only notes these two duels (if Musashi's victory at age 13 can be called a duel) without giving details.)