Good question. They are both phrasal verbs which have the same meaning (to take care of something): "to look after" - used to express taking care of an object or a living thing (i.e He looked after his neighbors cat when they went out of town. He also looked after the house. "to care for" - used to express taking care of a living thing (i.e She cared for her Mother because she was sick)

Look after: phrasal verb, which has two meanings: If you look after someone, it means you do what is necessary to keep them healthy, safe, or in good condition; If you look after something, it means either you do what is needed to keep them in good condition or you are responsible for it and deal with it or make sure it is all right, especially because it is your job to do so. e.g., Look after the children means taking care of the children. Look after the business means running the business. Care for: phrasal verb. Care for something means you want sth or you like sth; Care for somebody means take care of sb. or support sb. e.g., Care for red color means you like red color. Care for the old means you will support them when they are aging. So they have similar meannings when talking about taking care of sb. or caring for sb. Look after sb. means you have to do sth. to keep sb. in good condition. e.g., I have to look after my little sister when my parents go out. Care for sb. means looking after or support. e.g., Lily spent years caring for her sick uncle. Chinese authorities are trying to educate the youth about the need to care for their parents. Another point is that Care for is more frequently used in formal occasions, like formal speeches, official publications etc.

to me: when you look after someone, it means there's an action you have to carry out physically. whereas care for some1 or sth is merely a thought, feeling in your mind, which can of course lead to action, if you care enough for it. so in other words: look after = physical care for = mental