"care for" can imply action - you care for a baby or sick person by feeding them, making sure they're dressed and clean, etc. "Care about" refers more to emotion exclusively.

"She cares for her elderly mother" makes you think she lives with her mom and helps her with activities of daily living.

"She cares about her elderly mother" does not make you think that.

If one's t says either thing, best to go directly to the t for clarification/expansion.