Because there are several other factors than physical resemblance when it comes to casting in biopics. In fact show me a bankable actress from these days that looks like the woman on the picture you've posted because you know, that was a different time and therefore different fashion. When you really look for more pictures of Billie Jean King you can in fact tell that casting Stone is not more off then Nicole Kidman as Virginia Woolf, Charlize Theron as Eileen Wuornos, Philip Seymour Hoffmann as Truman Capote or Cate Blanchett as Katharine Hepburn. And guess what? They all won oscars for their roles.
Seeing the first pictures of Stone on set they clearly are going for the same physical transformation deglamming that already worked a dozen times in the hunt for Oscar buzz.
The movie attempted to present King as an attractive guy magnet who just happened to discover that she was a lesbian in her late twenties. In reality, she always appeared to be a lesbian, and not particularly attractive to men. That's not PC, but it's true.