“Sich kümmern um” and “sorgen für” referring to a person are often synonymous. “Ich sorge für meine Mutter.” means that I ensure my mother’s well-being, which is kind of the same as saying that I take care of my mother: “Ich kümmere mich um meine Mutter.” Notice that “sich kümmern um” is accusative-reflexive and takes “um” as a preposition, whereas “sorgen für” is just a normal intransitive verb (not a reflexive one) and it takes “für” as a preposition. Let me know whether that explains things. :)