"Delegated" carries with it the connotation that one is acting as a representative of a superior in carrying out a task. "Assigned" merely means told to do a task.
If I were your superior, I could delegate you to carry out one of my tasks, or assign you a task of your own level.
Assignment - the process of imposing responsibility to our subordinate, of the task we are not responsible.
Delegation - The process of imposing responsibility to our deputy/sub ordinate, of the task we are responsible and are represented.