They are same, 제= is the short of (저의) and it is a formal form , but 내(나의) is informal.

내 is casual. 제 is formal.

je is polite speech and ne is plain speech

Formality - 제 is the polite/honorific form

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.

Nae is informal and jae is formal

They are the same, 제 is formal while 내 is informal.