I would say there is, but it's difficult for me to pinpoint. It has to do with context.
"What do you want with me?" means roughly "Why are you seeking me out?", in my opinion.
"What do you want from me?" means roughly "What do you expect (to get) from me?", as I hear it.
If a girlfriend said, "I always thought I would end up with someone who was more successful", the two questions would convey very different moods.
"What do you want with me?" = "Why are you in a relationship with me, then?"
"What do you want from me?" = "What do you expect me to do (to fulfill these wishes of yours)?"
If a policeman comes to my place of work and asks to see me:
"What do you want with me?" = "Why are you asking to see me?"
"What do you want from me?" = "What do you expect me to give you?"
"What do you want from me?" can be belligerent or sarcastic, as in:
A: "I finished my homework."
B: "So, what do you want from me? A gold medal?"
You can't substitute "with" for "from" in this context and have it make any sense.