The two forms are similar. But there is a little difference.
When I say: "Faça isto por mim." (Do this for me.) This is an emotional appeal or a favor.
For example: The husband is leaving the wife, he no longer loves her. But he has a son, then his son request: Dad, don't go... for me. (Pai, não vá... por mim.) I mean, the husband no longer loves her, but he won't go because his son asked.
When I say: "Faça isto para mim." (Do this to me.) Is almost the same thing, but there is a subtle difference.
For instance: I'm tired, but it is my turn to take out the trash. So, I ask for my younger brother: Please, take out the trash to me. I'm very tired. (Por favor, tire o lixo para mim. Eu estou muito cansado.)