You can do both. Start a new game on Extreme without the weapon upgrades but the ability to save whenever you want OR replay missions from your original save file (with all weapon upgrades) and select extreme difficulty. The only downside of using your original save is that you can not save when you select replay mission. You do get checkpoints though and once you've played a level once you get to know what to do and can really run and gun with Lang. I just used my original gamesave as I felt to underpowered starting over. I had to go to carefully.
-Insert something creative here-
( that doesn't include stats that no one really cares about or some poorly photoshop'd image with your name stuck in it. )