Consign means to give something over to someone else’s care. So, let’s say you take some of your old clothes to a 2nd hand store for them to sell on your behalf. You have consigned the clothes to the store.
Assign can be used to mean basically the same thing as consign (to transfer), and can also be used in other ways. You can appoint someone to do something (I assigned the class a book report for homework) or to a position (he was assigned the job of Vice President of the corporation) or to attribute meaning to something (After thinking about what Bob did, Sue assigned a negative motivation to his actions).