In the Book
In the Movie
Mary Jackson was given an assignment by John Becker, the chief of the compressibility division. He challenged her work, she stood her ground; the problem was with Becker's input data.
This doesn't happen in the movie.
Two weeks after being hired in 1953, Katherine Johnson is sent to Bldg 1244 to work with engineers along with Erma Tynes. This is done at the request of one of the male engineers. Virginia Tucker, the actual Supervisor got along well with the colored computors.
She is sent to work in the engineer's office in 1961 at the request of Vivian Mitchel (composite character). She tells Katherine they’ve never had a colored person in that department and not to embarrass her.
Katherine Johnson and Erma Tynes had little trouble fitting in with the white male engineers.
Katherine is treated like a janitor at first and provided a “colored” coffee pot to use. Erma Tynes is not in the movie.
Katherine is never treated like this by the other engineers.
When Katherine is assigned to review another engineer's math results, he gives it to her with much of it redacted (blacked out). She holds it up to the light to see what he had blacked out (info on Mercury and Atlas projects) and after checking his work graphs some of it on the large chalkboard in the office.
This does not happen in the book. But during the red scares of the early 1950's, a black engineer was fired for being too radical.
After the engineers see that she gained access to information on Mercury and Atlas, she is sternly questioned in her boss' office about how she got the information; they suggest she might be a Russian spy. She denies it and explains that she simply held the papers up to the light to see what she needed to know.