פירוש על עבודה זרה 100:12

The baraita distinguishes between an idolater who uses stones from a Mercurius to pave the roads, in which case a Jew can walk on the roads, and a Jew who uses these stones, in which case it is prohibited for the Jew to walk on the roads. But Rabbah b. Yirmiyah cannot explain this distinction. “Carpenter” here seems to be a nickname for a scholar, probably due to the high level of skill a carpenter needs to shape his material.

Breaking the stick in front of the idol is considered an act of idolatry and thus if an Israelite does so he is liable. But throwing the stick at or in front of the idol is not considered an act of idolatry, and thus one who does so is exempt. But why the difference? After all, both have a certain amount of similarity to an act performed in the Temple. Breaking the stick is like slaughtering an animal, and throwing the stick is like sprinkling the blood.
Rava explains the difference. For an act to be like sprinkling blood it needs to be an act that is broken up into several parts. When one sprinkles blood on the altar, it is done in parts. Since the stick is thrown all at once, one is not liable.