Hmm, they essentially mean the same thing. I would use "to look after" more, if you are talking about a shorter period of time. Eg. I will look after your cat for today. Can you look after my keys? "To take care of", you can use more with people and for longer periods of time. It implies that you'll do more than just watch them. Eg. I'll take good care of your daughter while you are away. You can also "take care of" a problem, but you can't "look after" a problem. :)

They are almost always the same. I cant think of a situation that one works but not the other.

Creio que necessariamente é a mesma coisa