More teminology changing! Well here is my understanding.

A white supremacist is a person who believes that black people are biologically inferior and that therefore it is right and proper for whites to be in control and treat black people as inferiors (e.g. KKK etc)

White privilege is a term used to describe the priviledge in life (perhaps unbeknownst to them) that white people have due to many subtle and not so subtle behaviours of people and institutions.

A person with white privilege may desire to be behaving fairly, this is not true of the white supremacist.

So they are different.