To me, a really interesting difference between the Moulin Rouge movie musical and the Moulin Rouge Broadway musical is the degree of agency that Satine has... especially! In relation to Harold Ziegler and the dance hall itself.
In the Broadway show, their interactions carry a much more respectful, two-way energy. I don't know that I'd say she's fully his equal, but they work together as though she has a stake in the company. Not majority ownership, but like she helped found it or co-signed something; IDK if that's a possibility, but it's the energy that their dynamic carries.
It's not that he's telling her to seduce someone, it's that they're planning it together. He lies to her about some things, yes, but he doesn't order her to keep stringing along the Duke. It's more duplicitous, but in that duplicity, the show carries the message that he believes she has too much agency and independence for him to control her outright the way he does in the movie, and that's why he has to lie about the contract he signed with the Duke.
I need to be clear that in the stage version, part of why Satine is doing this shit is because the dance hall is going under. Harold is cutting staff left and right to make rent. Fewer servers, just fired the last sommelier, that sort of thing. It's why he urges Satine to seduce the Duke; he's trying to save the dance hall by getting a new investor, and she knows that.
Satine is also the only one that he's actually honest with about the dire financial straits. Everyone suspects it, but she's the one that he actually lets in on the issue, and these factors are part of what sells that more equal dynamic that I mentioned. He's not her pimp, he's her co-founder, but since Harold can't get that funding himself Satine does it.
An explicit comment in the show is that without that funding, the dance hall goes under, and they all end up back on the streets. Satine seducing the Duke is her making a choice between one guy, or having to switch to full-service sex work full time again, especially since she's not getting any younger, and she's unlikely to find more work as a performer at her age.
"Some more rouge, perhaps?" kills me because like. He treats her like an equal. She has denied his offer to step down, to heal, and let someone else take the center role until she's feeling better.
The Broadway show is also a lot less weird about the topic of virginity, and also sex work in general. It's pretty clearly indicated during "Sympathy for the Duke" that they boink that very night, and regularly afterwards. His motivations aren't just about having her purity or whatever. He wants genuine affection. He doesn't promise to marry her or anything, but his intent is to have her move into his house, and he says "all of this can be yours."
He is very selfish about wanting her love, but it's all about her heart, about her affection and time and attention, and not just her body. The movie's "Like a Virgin" is funny, but the Broadway show's choice hits so much harder thematically.
He is also much more of a threat on a personal level than just having minions. Every Duke I've seen on Broadway has carried an air of menace that suggests he can and will do actual gangster-style violence to people who cross him.
Also the way that Satine and Toulouse are written in relation to each other is great. (I will never be able to see that man as anyone but Andre Ward. He is phenomenal.)
(For anyone who needs some context here, I have seen this show live three times. I kept winning discount tickets and had multiple family/friends who wanted to see it, so I just took a different person every time since you can only win two tickets, and the person who won has to be one of the two.)
ALSO, the Raise a Glass number! It's so fun! Paired with the specific way that she tries to seduce Christian later, it sells the sudden romance a bit better to me than the movie does.
Her relationship with Nini is also a lot more nuanced. Nini is both ferociously jealous of her and protective. When Satine starts coughing and nearly passes out in the middle of her big opening number, Nini runs to her to help her get back to her feet before the clientele notice. When Satine and Christian are fucking around behind closed doors, Nini is bitchy about it, but she warns Satine that hey, you're gonna get caught. She's not nice about it, she's even a bit slyly threatening about it, but she warns her instead of selling her out in order to get the part. When she sees blood on the handkerchief, she's concerned and she does her best to help Satine hide the issue. When Satine is almost too sick to perform, she does offer to take on the role as understudy, but she's very obviously not happy about it, despite earlier having said that she'd do anything to have Satine's headlining role.
And on top of all that!
"Spectacular Spectacular" isn't intensely racist in that very victorian way anymore.