They are same, 제= is the short of (저의) and it is a formal form , but 내(나의) is informal.
Politeness! :) 제 is more formal, lowering yourself before the other person. 내 is relaxed and more used, in my opinion, but thats because I'm used to hearing my elders talking to me or listening to friends amongst themselves.
heavy audible sigh my habit of saying bookbag instead of backpack cost me the right answer :D