"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.

https://english.stackexchange.com/questions/6278/delegated-to-vs-assigned-to

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.

https://www.bayt.com/en/specialties/q/202836/what-is-the-difference-between-assignment-and-delegation/