If I say "take care of a person" or "look after a person", it usually means the same thing, to provided care over a period for a person. It is not always true, however. "Take care of a person" can also be used to mean providing a resolution to a person's request or problem. It implies a short term solution. "This customer has a request concerning the new computer. Could you please take care of them." In this case the person being spoken to is being asked to give advice or short-term help to the cusotmer, not provide care. Sometimes you will hear "look after" in this regard, but not as often. You would also "take care" (resolve) a problem. Again, there is considerable overlap here, so you would not be guilty of some huge error if you interchanged the two,.

Take care is used when you are saying goodbye to someone. It actually means "Take care of yourself." for example: "Bye/See you later. Take care. " Take care of means to look after someone or something. for example: "You should take care of your new car, it cost a lot of money."