Meaning: It is used to refer to someone who is not violent at all, who is harmless:
Do not worry about that lioness. She would not harm a fly. She is my friend.
I do not think that Mr Evans is capable of doing such a thing. He would not hurt a fly.
Meaning: The use of brute force, or the act of threatening someone or something weaker.
The police used the big stick to reduce the demonstrators.
That ruler seemed to be a nice person, but carried a big stick.
There are politicians that wave the big stick in the name of democracy.