"Ask after" is used for inquiring about someone's health or well-being. In the same instance "ask for" can also be used.
"Ask about" is generally inquiring for someone, their whereabouts, reason of absence etc. It can also be used to ask about one's health but then the word "health" will have to mentioned. Whereas "ask after" and "ask for" are inclusive of it.
He asked after/for you. (about your health especially)
He asked about you. (where you werr, what you were doing these days, etc)
He asked about your health/ how you were feeling/ well-being.
There is another phrase "ask around about".
We shall ask around about him. (We shall ask different people/ sources about him.)
This was my explanation. Obviously if there are any "mistakes" in my understanding, other competent posters will indicate so.
NOT A TEACHER
Student or Learner