Why would he do this to me?

They could have the same meaning depending on the context. The second sentence is in the simple past. He did it; it's done; you are left wondering why he did it.

The first sentence, could mean he did it and now you're wondering why he would have done such a thing. However, the first sentence could also mean that you anticipate that he is going to do something to you (the this) and you are wondering why he would.

If it's clear from what precedes the sentence that he has already done something, then I think these two sentences have the same meaning.

In practice, they are more or less interchangeable. If asked to define a difference, I would say:

• The first carries the idea that something unbelievable has happened (My neighbor just blew up my car; why would he do this to me?)
• The second carries the idea that something more normal, yet still unexplained, has occurred (My neighbor's dog just bit my leg; why did he do this to me?).'

I would guess that the first is more common. (Google confirms, with twice as many results for the first as for the second.)

Sorry, I'm not a native English speaker, but I would say there is at least one difference: "Why did he do this to me?" implies for sure that the action already took place, whereas "Why would he do this to me?" might refer to some action that has not happened, yet (and, possibly, will never happen).