Movies always leave out important details, they kind of have to or the movie would be hours and hours long. Stilgar doesn't want to kill Paul, because Kynes instructed the Fremen to find him. It is Jessica he would have killed, because he thought her lack of knowledge about the desert would put them all in danger. Then she proved herself and he realized she was a Sayyadina (a priestess who hadn't drank and converted the Water of Life). Plus the Bene Gesserit had planted mythology in the Fremen culture in preparation for this kind of emergency. Jessica quickly realizes this when they say certain things. She knew she needed to best Stilgar but not kill him.
There were lots of things the movie left out, that I wish they'd kept in, but I thought they did a pretty good job. What I missed the most in the movie was the sort of mythology that the books invoked. I also don't think they really conveyed how brutal the Harkonnens had already been to the Fremen, which would have made their hostility towards outsiders more understandable.
(lol I know I am obsessed with Dune)