It's basically the same game, but the Director's Cut adds some stuff, like voiceovers for the in-game characters (which isn't in the original game). This adds more context to the backstory of the game, since in the original nothing was explained.
Also, there's new speedrun levels and a new soundtrack. And if I recall correctly, the Director's Cut also supports Oculus Rift, which the original doesn't.

Which one you play (first) is up to you. If you like the mystery aspect more, play the original. If you want more story, play the Director's Cut. And of course, if you want to play using the Oculus Rift, you'll have to go with the Director's Cut.