Care about vs. Care for

Care about- to think that something is important and to feel interested in it or upset about it:

Care about- to be interested in something, or to be anxious or upset about something:

Care for- used in polite offers and suggestions

Care for- to provide for a person or animal’s needs and to protect that person or animal [= take care of]

Care for- to protect someone or something and provide the things they need, especially someone who is young, old, or ill [= take care of]

Care for- to love someone and feel romantic towards them:

Care for- to like something or someone

Care for- to take care of someone or something [= take care of]

Source:

Cambridge dictionary