The Director's Cut has a bunch of changes to the original game.
1. The DLC(Missing Link) was integrated into the game at the appropriate time in the story. You can't skip it. It may be annoying. Sorry.
2. The completed achievements do not carry over from your other game. It's treated as a completely separate game in Steam. All the DX:HR and Missing Link DLC Achievements were included.
3. There were some pre-order bonuses and collector's edition bonuses that gave extra weapons and credits. These were moved to the first mission (sarif manufacturing plant), and can be found early on in the mission, as opposed to just given to you at the start. Unfortunately the extra credits don't seem to exist anymore :/
4. The developer's commentary in the game can be found in the 'extras' section. It defaults to off, but you can turn it on and off at will during the game. If you accidentally trigger a developer's commentary and you don't want to sit through their sometimes rambling dialogs about various things in the game, just disabling the commentary in the menu, going back to the game, and then re-enabling the commentary, will reset it. This can be important, since some developer commentary will prevent certain actions, like talking to NPCs, while it is talking.
5. The boss fights were significantly overhauled and improved. While you still have to kill the boss, there are a lot of options on how you can do it. You can do things like hacking and stealth to use the environment against them. A lot more fun than just run and gun.
6. There is a new feature called 'new game plus' which is, just as you probably imagined, allows you to play a new game with all the augs bought from your previous game. No credits or items will carry over, but heck, having a fully auged out Jensen is good enough, right? It makes the early parts of the game comical and a lot of fun.
