In BBC, Irene is hyper-sexualized, introduced as a lesbian woman, who’s sexuality is then ignored in order for her to get wet in the panties for Sherlock. It is revealed that she only posed a threat in the first place because she was given help from Moriarty, a man. She is then defeated when Sherlock proclaims that sentiment is a “chemical defect of the losing side", reenforcing "Women are so weak and emotional and girly feelings make you weak" tropes, and the episode ends with Sherlock having to go and save her from the scary scary terrorists.
In Elementary, however, Irene works to SUBVERT the classic “damsel" trope. We are told that she was killed to get to Sherlock, only to be recovered and in need of “healing." (Bonus! Even in scenes where she is changing, she is never hyper-sexualized) However, the entire thing is a trick. Irene is actually the one in power. She manipulated SHERLOCK’S feelings in order to dominate him. Her needing rescue was only a trick, praying on Sherlock’s emotional buy in to the classic “damsel" narrative. The vast majority of the male antagonists in the show were only HER pawns. And let’s be clear here: She DOES beat him. She defeats Sherlock. Irene/Moriarty is only defeated because she underestimated JOAN, not because of her sentimentality. Because she only saw Joan as a “mascot." (And woah, the racial implications there are intense). In fact, Moriarty is primarily empowered by her emotions, talking about how her ability to manipulate people’s feelings is one of her sources of strength.
You see, the BBC narrative buys into sexist tropes and the systematic disempowerment of women, along with a healthy dash of queer-baiting.
Elementary deliberately subverted those sexist tropes, demonstrates the power of women, and only has Moriarty defeated by underestimating another woman.