These two expressions can be a little bit confusing. First, let us remember that "care for" means "look after" or "take care of", as in:
1) She cares for her sick father.
This meaning of "care for" I will not discuss in this Column.
"Care for" is almost always used with a negative ("do not care for"), and in such cases means "do not like". Here are some example sentencs:
1) He is a man I do not care for.
2) I do not care for hot weather.
3) I did not care for my job, so I left my company.
"Care about" is usually used with a negative, and in such cases means "do not have any interest in", or "be not worried about".
4) I do not care about politics.
5) I am not listening to you because I do not care about your opinion.
6) There is nothing he cares about more than himself.
However, "care about" can also be used in an affirmative sentence.
7) I do not love you, but I care about you.
8) We should start to care about global warming.
9) You should start to care about your health.
Among sentences (4) to (9), sentences (8) and (9) seem to have a stronger meaning of "be worried about".
To contrast these two expessions, we can put them in similar sentences:
10) I did not care for my job, so I left my company.
= I did not like my job. / I hated my job.
11) I did not care about my job, so I left my company.
= I was not interested in my job. / I had no interest in doing my job properly.
an affirmative sentence：肯定文