Porting: Making an entirely new game that looks, plays and feels like the old game. Important part: You need to make every game from scratch (yes, it is oversimplified), which usually means several months worth of work for one game.
Emulation: You make the machine pretend to be the machine the game was originally made for. Lots of effort up-front, extremely inefficient performance-wise, but you can pretty much reuse the original games as they were, so doing the same procedure for a new game simply involves taking the same emulator and putting in another game.
You cannot just swap the game in a port, because the original game is nowhere in there - the port is an (almost) entirely new piece of software written for the new machine from the ground up.