The main difference between the two is the characters; the characters in the film adaptation are different than the characters in the short story.
Also, both take place after World War II but the story's time frame appears to be soon after the end of the Second World War. So it took place sometime after 1945. The film adaptation's time frame is the 1960s. The short story is set in England whereas the film takes place in the US.
Perhaps the biggest difference between the story and the film is that Alfred Hitchcock shot the film in the little village of Bodega Bay, which is just a little bit north of San Francisco, whereas the story by Daphne du Maurier is set in Cornwall, England. The Hitchcock film features American actors Rod Taylor and Tippi Hedren. The short story in a lot more gruesome than the film because the birds in the original story cause vastly more death and destruction than they do in the Hitchcock film.
As the link to "The Birds" on enotes points out, the short story by DuMaurier "presents an unrelenting portrait of terror and a compelling analogy of the atmosphere of fear generated in America and Europe during the Cold War years." Although it concerns a family, it does not have the love story that is central to the Hitchcock film. Also, the film emphasizes nature's unexplained rebellion against humans, suggesting a revenge theme, while the short story, as eNotes points out, was written to depict the horror of the bombing in England during WWII and fear of a nuclear holocaust in the 1950s and 60s.