Director's Cut observations:
Episode 23'
Rei... She is the product of my despair, and at the same time... the vessel of your hope even now.
There are a number of very significant differences between the original version and the Director's Cut of this episode. The main ones involve Rei's contact with the 16th Angel. Firstly, there's the image of the tower of Angels that burst out of the back of Unit-00, separating its armour. Also, as the Angel tries to attack Unit-01, it becomes a vision of Rei, shrieking in her voice as Shinji stabs it with the prog knife. Shinji's hand then bubbles up into a cluster of mumbling Rei effigies.
Then there's an image even more unsettling than the tower of Angels: the belly of Unit-00 bloats up in a grotesque depiction of pregnancy as it absorbs the Angel into itself. Note also that the Angel is absorbed between the Eva's legs. The image of the belly collapsing back in on itself as if crushed by extreme pressure is something that would later be used by Gurren Lagann in a particularly impactful scene of its own.
Finally, there's one of the most memorable images of the series for me as Unit-00 morphs into an Eva-sized vision of Rei, complete with halo. There are sounds of ecstasy heard just before it self-destructs. Both visions of the glowing white Rei pre-empt its appearance in The End of Evangelion.
Beyond this, Rei's facial expressions in her scene back at home have been redrawn
brilliantly. Whereas before her expressions were completely blank and unreadable, she now looks tired and fed up while looking in the mirror, and coldly resentful while looking at Gendo's glasses. Ritsuko's expressions have been revised, too. in the original episode, her face as she descends the escalator in NERV HQ following her release by SEELE is completely static; in the revised version it communicates some kind of internal conflict that she seems to resolve and reach a decision on. And that decision involves the Dummy Plug plant.
Smaller changes include new on-screen displays on the monitors at NERV, and a honeycomb background behind the lift as Ritsuko, Misato and Shinji head downwards inside.
Also, and I didn't pick up on this myself, partly because the terminology seems to be a bit off, but on the walls of Rei's birthplace are scrawled the words "top", "bottom" and "strangeness". These are references to quantum mechanics, as explained here:
wiki.evageeks.org
I personally find the article overall to be another example of overanalysis, and would instead suggest that the presence of the words perhaps simply offers some kind of deliberately vague "explanation" as to how Rei was cloned in the first place.
Episode 24'
"Humans forgot their foolishness and repeat their mistakes."
"If humans do not redeem themselves willingly, they will not change."
As well as the brand-new conversation scene with Kaworu and the monoliths at the lake, SEELE's dialogue as he descends towards Terminal Dogma has been changed entirely. This is the only dialogue change between versions, so it must surely be significant. Directly from the ADV subtitles, then...
Original:
"The last Angel has invaded Central Dogma. It is currently descending."
"Just as we planned."
"Ikari, you were our good friend, our comrade and conspirator, but this is your last job. I expect that you will use Unit-01 to accomplish the mission."
Director's Cut:
"Humans forget their foolishness and repeat their mistakes."
"If humans do not redeem themselves willingly, they will not change."
"We will not rely on the power of Adam or the Angels."
"Our only choice is to make changes with our own hands for the future. I will be praying that Unit-01 will accomplish its task."
Very much different in its focus.
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For the sake of completeness, here's a set of links to Evageeks' studies of the differences between the original On Air and revised Director's Cut versions of episodes 21–24:
Episode 21:
https://wiki.evageeks.org/Guides:Episode_21_OA_vs._DC
Episode 22:
https://wiki.evageeks.org/Guides:Episode_22_OA_vs._DC
Episode 23:
https://wiki.evageeks.org/Guides:Episode_23_OA_vs._DC
Episode 24:
https://wiki.evageeks.org/Guides:Episode_24_OA_vs._DC
List of all the comparison pages on the site:
https://wiki.evageeks.org/Category:Comparison_Pages
@~AyaMachi~: The difference in Rei's expressions is really worth a look. It's in the Episode 23 link, further down the page. It starts with the cut that's numbered 246 in the left-hand column.