The beginning of the two movies is slightly different. The comparison is based in the structure of the Theatrical Version. Here a little overview.
Joe can't have an orgasm anymore and the audience gets more information about Seligman who considers himself asexual. Furthermore, Jerôme and Joe are having a child, they are trying to live a family life.
Director's Cut 8:16 minutes longer in total
Extended shot of old Joe describing the two women in her vision from when she was 12 and had her first spontanious orgasm. Seligman is obviously upset about that story. In both versions, he calls it a blaspemic re-narration of the glorification of Jesus on Mount Tabor, one of the holiest bible passages of the Eastern Church, in which the humanity of Jesus Christ is being enlightened by the devine, eternal light. But it becomes clear pretty fast that one of the women was Valeria Messalina, an infamous nymphomaniac and wife of the Roman emperor laudius and the other one was nobody else but thw Whore of Babylon. All that is in the Theatrical Version as well but more summarized and without Seligman's anger being so obvious.
Then Seligman says in the Director's Cut that Joe demanded a lot from her audience and Joe replies that she was as innocent in questions of faith as Seligman was when it came to sex. He smiles.
Alternate angle when Joe is trying to have an orgasm under any circumstances (please see above).
Seligman's mathematical explanation (the race of Achilles and the turtle) why Joe can't have an orgasm anymore is longer and more graphic in the Director's Cut.
Seligman then elaborates the meaning of icones in the orthodox church.
Subsequently, the Director's Cut only contains a flashback of Joe hoping to regain her sexuality. She is walking across the park and notices 3 leaves dancing in the wind. That causes her need to violently get her sexuality back.
Then she has sex with Jerôme.
A part of this sex scene later in the Theatrical Version during another scene
The subsequent restaurant scene - Jerôme offers her money to jam an ice cream spoon up her vagina - lacks short moments. For starters, she spreads the ice cream spoons on the table longer plus another couple watching.
Then a short missing scene during which Jerôme takes a closer look at Joe's genital area - he still seems to be surprised about the huge number of spoons she put inside of her.
After Joe had her child, the sex scene is more explicit in the Director's Cut when Jerôme can't get it up. As compensation, the Theatrical Version contains a part of an earlier scene - please see above - which makes the Theatrical Version longer.
A scene of the family life. Joe expains from the off that she had never mentioned sex with other men to Jerôme, even though it was him who suggesed it in the the first place.
Nonetheless, he has trouble handling it and sends her empty letters with the name of unknown male names in order to test her because the arrangement is a real burden for him. Woul she show him these "love letters from strange men"? She does not. And that only makes him more jealous.
3 Years later: Joe is now being played by Charlotte Gainsbourg in the flashbacks as well. Joe is looking for new sexual experiences. Among other things, with men she can't communicate with because it is all about the sex.
Director's Cut 113 seconds longer in total
The black guys in front of Joe's window, the conversation with the interpretator and his persuading that one of them should have se with Joe takes more time in the Director's Cut. Nothing relevant is revealed though.
When she then has sex with the black guy and his brother, the double penetration is much more graphic.
During the subsequent conversation with Seligman, she also talks about the burning of books when Seligman criticizes her for using the word "nigger". She does not want to get censored.
When the issue is whether or not humans deserve democracy - she does not think so - she also says that the society is based on hatred when it should be based on foregiveness.
Joe tries to test the boundaries and meets the sadist K. The sessions at night with him lead her to turn her back on her son Marcel. In the end Jerôme takes Marcel and leaves her.
Director's Cut 14:18 minutes longer in total
Because K. is convinced his work was not interesting for Joe but she is persistent so he demands a test from her in the waiting room. When he strikes out, she ducks and screams "No". K. lets off of her and leaves the room with the other woman.
Later on, Joe returns and sits on the chair in the middle of the waiting room, willing to be testes again. K. hits her twice. Then he accepts her as a client.
In the Theatrical Version, these two scenes have been combined to a single one. That means Joe does not blow it at first (no pun intended :-) ). Instead, she makes the test during her second visit and passes immediately.
Joe, who is called Fido by K., buys the riding crop in a single line store and not in some cheap SM store. K. insisted on something authentic.
After that, she drives to K.
K. wants to tie her up but he does not at first. He waits until she is already on the couch.
When he says her butt was not high enough, he interrupts the session but he grabs her vagina to test the moisture. More graphic in the Director's Cut.
During the following conversation with Seligman who just does not get it, Joe explains she was wetter during the second session which also explains why K. eagerly hit her. If it was really the books underneath her belly so that she was lying higher remains unclear.
Then she compares herself to a pot plant. Just like one has to check if the plant is still wet enough, K. did the same thing when he was hitting her.
The next session with K.: next time, he wants Joe to take 15 small coins. Then one can see her collect the coins at home a couple of days later. She says goodbye to the child and drives to K. The child is obviously home alone, with no babysitter.
During a subsequent session he also shows her to make the blood knot for the cat o' nine tails. He even gets out a folding meter stick in order to get the correct distances. Then he eplains it was the knots that tear the skin open while being wipped. Meanwhile, she keeps working on the cat and he keeps philosophizing about the knots including the slipknot.
After the last session with K., there is more footage with Seligman and Joe. First, Seligman explains he always expected Joe being gone when he walked into the kitchen in order to make fresh tea.
When K. then teaches Joe the Silent Duck (he shoves his hand up her vagina), the Director's Cut is more explicit again.
Joe changes the subject with Seligman. Enough of K., the sadist. Now, she speaks of a prostitute who told her that masochists were the toughest people to satisfy and they were never grateful, different from the others.
Then Joe turned her back on sadism and masochism but she had regained the capabilitly of having an orgasm.
A shot of her throwing the cat in the lake.
Seligman still does not see anything evil in her actions which is why she brings up the story of younger Joe who is still in a relationship with Jerôme.
He once gave her a ring and she made a game out of it: "Cinderella". He may only look at the ring but never touch it and he happily played along. Then she throws the ring in the fireplace and dances across the room while singing "Cinderella, Cinderella ...". Jerôme tries to extinguish the fire and get the ring back. He does not enjoy it.
He returned the ring to the jeweler and got a refund.
Old Joe asks Seligman if he considered that evil but he does not know how to respond. Since she is wearing a diamont around her neck at the very moment - a gift by some man she can't remember - he starts another digression. About diamonds and boats, especially about the meaning of the cut of brilliants.