In the original Resident Evil 2, Leon and Claire went through different paths and played a different game from each other, sort of like Jill and chris in RE1, or to make a genre switch comparison, Sonic and Knuckles in Sonic 3. Or like the difference between the first quest and the second quest in Zelda. They go through a lot of the same places, but the actual content of the game is different and remixed. They also fill in parts of the story, the story doesn't make full sense until you played both halves.
I haven't played the new one, but it sounds like after beating the game you can play as either character in either scenario. So what was originally Leon's game, can now be played with Claire, and vice versa. But there is also an actual second scenario, that originally played with Claire, that he missed.
Okay, so basically, once you beat the game once it's back to the way the original version was, where you can play Leon A, Leon B, Claire A, and Claire B? I don't remember if all those were immediately selectable upon starting original RE2.