The main differences I have noted are the amount of batteries you can carry and the hits you shall endure before dying.
The highest difficulty mode (Insane)is a "YOLO" I think, with saving disabled and NO checkpoints. I finished the main game on normal a long time ago and I had 10 batteries. Recently, after grabbing the DLC, I am playing on the second most difficult level (Nightmare) and I can carry only up to 2 batteries. The monsters seem to be as dumb as in normal, but they are able to kill you faster.
Personally, I wouldn't try Insane, as having to replay the entire game for a mistake and having to finish it on one run, kills the fun for me in these games. I think Nightmare is a good level because you have limited batteries, and you you must save them, so you will have your cam's nightvision off more time, creating a scarier game. Nevertheless, you can get still enough batteries to play the game and not having trouble because of not seeing anything.
I shall paste also info from the wiki
for a more precise answer:
Initially, Outlast only had a standard difficulty. After its update, the game features three standard difficulties; Normal, Hard and Nightmare. When playing on the regular difficulty, enemies are less aware of the player's whereabouts. As the difficulty increases, the door shut sound becomes more loud, The Variants become stronger and wiser, being able to kill Miles very easily and making it difficult for the player to hide, as they thoroughly search rooms and various hiding spots. In addition to all of the previously mentioned disadvantages, the Nightmare difficulty limits the player to a maximum capacity of only 2 batteries. After the sixth update on Steam, players who download Outlast from torrents are able to play in Hard and Nightmare difficulties. In addition to all of these difficulties, the PlayStation 4 and Xbox One versions of the game contain an Insane difficulty. It has neither checkpoints nor saves, meaning that if the player dies, they will lose all of their progress and be forced to start from the beginning of the game. After the April 30th update, Steam version of the game also has the previously exclusive Insane difficulty.