People who are “appointed” for a job are chosen by others for a particular position. An appointment is something official.
* The members of the Board appointed Dravid captain.
“Anoint” has a religious connotation which “appoint” doesn’t. When a priest anoints someone, he applies oil or water on some part of the individual; this is usually done as part of a religious ceremony.
* The young priest anointed the old man’s forehead with sandalwood paste.
It is also possible to “anoint” oneself with something. In this case, we are merely smearing ourselves with some sort of liquid. For example, an individual can anoint herself with suntan lotion. When a person in authority, such as a priest, “anoints” someone, he chooses this individual for an important job.
* The Archbishop anointed her queen. In this example, the Archbishop is making the individual holy by anointing. In other words, he is consecrating the individual to office.
* The former Prime Minister did not anoint anyone as his successor.