They can be synonyms, for example: "I take care of the children" and "I look after the children". The expression "Take care" can also be used when you leave someone and you wish them to be well. But taking care can also be used to say "be careful" with something. For example, "Take care when you go hiking this weekend". And it can also mean that someone will take the responsibility of doing something. For example, "I will take care of this task". Hope this helps! :)

These phrases are considered quite formal. They basically mean the same thing. Usually when friends are ending a meet-up or signing off in a letter/message, they will say "take care!" which is an informal way of saying "please take care of yourself". However, they will not say "look after!" as there is no such phrase. Hope this helps! :)