Nobody has replied yet so I'll give it a try. I'm not a native English speaker as you know, so I may have some misconceptions about this.
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"By" introduces an agent, someone or something that, according to our perception, performs the action. "With" introduces an instrument, something that serves the agent to perform the action.
I was hit by my brother with a stick. My brother hits so he is the agent. He uses a stick, which is therefore an instument.
It's not a very common structure though. Often, when you use the passive voice, you do that because you want to omit the agent.
I was hit with a stick. We don't know the agent but we do know that a stick was used to hit you.
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The agent does not have to be a person.
I was hit by a car. Here, "a car" denotes the agent, not the instrument, even though a car is not a person. You may wonder whether cars are really capable of performing actions. Well, I don't know actually, but that's just how the English language sees it.
You're not very likely to say,
I was hit by a stick.
We don't think of a stick as of the agent of hitting. Why can a car be an agent and a stick can't? I think that's because cars seem to have a will of their own and sticks don't. Do you know the etymology of the word "automobile"?
