What will actually change if I decide to change my surname to my mentor's? I want to do so, but I don't want to miss out on any kind of exclusive backstory stuff if I don't keep my original surname. 

It depends! Characters who know your mentor or know your original family name may have unique things to say or treat you differently depending on what your surname is and how they identify you. There are different story things for both. My advice would be to go with what makes sense for your character - are they attached to the Order and the new identity they have, or do they want to keep some kind of connection to where they came from alive?