A person gets appointed to a job role (manager, administrator etc) another person is assigned a task. Or to put it another way; As soon as, "Paul" was appointed to his new role as, "General Manager" he began to assign tasks for his staff.

Since there is more than 1 meaning for these 2 words, here are the dictionary definitions: Appoint 1. To select or designate to fill an office or a position: appointed her the chief operating officer of the company. 2. To fix or set by authority or by mutual agreement: will appoint a date for the examination. 3. To furnish; equip: a house that is comfortably appointed. 4. Law To direct the disposition of (property) to a person or persons in exercise of a power granted for this purpose by a preceding deed. Assign 1. To set apart for a particular purpose; designate: assigned a day for the inspection. 2. To select for a duty or office; appoint: firefighters assigned to the city's industrial park. 3. To give out as a task; allot: assigned homework to the class. 4. To ascribe; attribute: sorted the rocks by assigning them to different categories. See Synonyms at attribute. 5. Law To transfer (property, rights, or interests) from one to another. 6. To place (a person or a military unit) under a specific command.

Appoint has some sense of honor. You were chosen out of a group for the role. Assign is less honored. You were probably going to get assigned something. Almost everyone will get assigned something... Here it is; your role... Big deal.