I don't get this whole "Vickers is a robot" thing. Why would anyone even think that? Because of Ridley Scott and Blade Runner? People seem to think that she's even "more advanced than David", although David was specifically designed to emulate human emotions, to make him more human and easier for humans to work with. Vickers is the opposite.. she's less human than David. If they were trying to be all sneaky with android Vickers, wouldn't it have made sense for her to be more human and not less?
The fact that Janek asks the question was probably a wink to the fans about the Blade Runner controversy.
But I think it's far more likely that Vickers is simply a human that doesn't know how to be human. She probably spent very little time with Weyland growing up. Knowing Weyland, he probably had various models of android watching over her, so that was her role model. As others have mentioned, Weyland is far more interested in his "son" David than his own flesh and blood daughter, so it seems like she would want to emulate the things he loves in the hope of receiving some of that love. It's quite clear in the film that Weyland doesn't give a damn about her or what she wants, and more importantly, he shows her none of the affection he shows David. If he loves his artificial creations so much (and he clearly does), why wouldn't he show the same level of affection for Vickers as he does for David? Especially if she were the "more advanced" version?
My take:
There is obviously a struggle/tension between Weyland and his daughter. Unless he had her when he was 60+ years old, the age difference shown between he and her doesn't make any sense. She should be significantly older.
So two possibilities:
What does make sense is that the real Vickers is just as smart as he is, and much older, and she created her own "David" only hers was better. She gave it her own personality and directives, made in her younger image, able to carry out her commands while keeping herself safe back on earth or secluded somewhere, running the company via the puppet.
The "are you a robot" question would make sense to trigger a failsafe "prove you're not a robot" response from the android. In the case of Janek's question - maybe he knew she was, and figured he had nothing to lose by approaching the sex question. As ship's captain, he probably was privy to a lot more information about what was on the ship than he let on. Would he want to risk losing his command by propositioning the boss's daughter, or his direct boss? Not likely. But if he knew this Vickers here was an android, he may be able to get away with it.
Vickers may or may not have been stronger than David. I tend to think she was, and this intimidated him. David may also have been familiar with android Vickers' agenda or had prior experience with the real Vickers using android Vickers to achieve her agenda, separate and in competition with her father's agenda. Either way, there is tension between David as the loved son and Meredith as the rejected daughter.
So in this scenario, Weyland is not interested in android Vickers because he knows it's an android, and know his daughter's directives to this machine are at odds with his goals. For whatever reason, he can't reveal that this Vickers isn't really Vickers - probably some legal thing.
Another scenario possibility: his real daughter is dead, and he pulled an A.I. - trying to recreate his daughter in android form. He tried to put her personality into it, and her attitude and emotions. But instead he created some sort of Frankenstein's monster, and realizes it's not really her - just some dark version of her that opposes him. But he can't bring himself to destroy it because he's already lost the real Meredith. He rejects her.
Whereas David is crafted completely differently, to be subservient and pleasant, and created from scratch - David can be the son he wanted.
Lots of possible themes to delve into here.
And isn't that what a good movie should do?> Engage our imagination with possibilities?
So i'll enjoy the possibility that androids are a bigger part of this picture than we know at the moment for a fact. They obviously are a big recurring theme in most of the Alien movies, along with genetically engineered life.