What does Nick mean when he says Gatsby took Daisy under false pretences and in this sense, how does he differ from Tom?
It is Nick who says that Gatsby took Daisy under false pretenses. Gatsby made Daisy think that he could take care of her because he had money and security.
He had deliberately given Daisy a sense of security; he let her believe that he was a person from much the same stratum as herself--that he was fully able to take care of her
According to Nick, Gatsby knew exactly what he was doing in making Daisy believe he could take care of her. It wasn't a misconception that he just let go; he purposefully made her feel this way because he knew it was the only way he could have her. He did this because he didn't expect to want to be with her forever.
He had intended, probably, to take what he could and go--but now he found that he had committed himself to the following of a grail.
Gatsby's original intention with Daisy was just to have a "fling" with her, so it didn't matter what she thought of him. Therefore, when he realized he was in love with her and wanted to be with her, he knew that he had to change himself in some way to be worthy of her.
This differs from Tom because Tom never put up any pretenses. Daisy knew exactly what she was getting into with him. Even with his affair with Myrtle, there aren't many secrets. Daisy knows that Tom isn't faithful, but it almost doesn't matter because he is able to provide the social status and security that Gatsby was not able to provide.
Gatsby ‘took’ Daisy under false pretenses; what were those false pretenses? In other words, what had Gatsby allowed Daisy to assume about him, a misconception that he does not correct?
He made her believe that he was in “the same strata” as she was and that he would have been able to provide for her financially when in fact he was penniless.
And, really, by not correcting her misconception he was letting her continue to believe it for his own gain.
Tom was born with money and privilege while Gatsby was broke but engaged in shady business dealings to earn his fortune. Also, Tom has no direction or purpose in his life while Gatsby has the goal to win back Daisy -- which is somewhat the reason he is not correcting her incorrect assumption.