To me, the biggest difference is of course that it is a musical. The songs allow the viewer to get a deeper insight into the characters' minds and hearts.
Firstly, Higgins song about letting a woman in his life confirms the theory that he is a misogynist who doesn't deal well with women. He cannot truly understand them.
Secondly, When Eliza sings "Show Me" we can see that she knows Higgins feels something for her and she is fed up with the men in her life who are tentative when it comes to expressing emotions of the heart.
There are many more but a scene that isn't in the play that I really liked because it said a lot about Higgins is the scene where the two men call the police after Eliza has disappeared. It's quite telling that the man who claimed not to care for any woman and who was quite careless about details knew Eliza's description better than the gentleman.
The most obvious difference is that My Fair Lady is a musical. (Always start with the most basic/obvious.) This creates several other differences. First, it is more markedly stylized; it is less realistic. Second, the singing people of London (or specific settings) mean that the community is more fully involved than in Shaw's play, where the core characters can act in relative isolation. Third, the singing changes the mood; it is much lighter at times, and more obvious throughout.
One difference is that in the play Pygmalion, Eliza enters the room when her father is finishing his busines of "letting her go" for a sum of five pounds. She is dressed in a Japanese kimono and Doolittle does not recognize her. In the musical, Eliza is cleaned up and dressed in Victorian clothing, not a kimono.
The ballroom scene is an "optional" scene which Shaw wrote for the screen version. Here, Pickering and Higgins are worried that the Hungarian interpreter (Nepommuck) and former student of Higgins' will discover their game. Eliza wins him over with flying colors since she is proclaimed to be royalty of some nature. This scene is not in the original play.
There is another "optional" scene in Act four when Eliza storms angrily out of Higgins' house since he has failed to recognize her part in his successes with Nepommuck. She runs into Freddy, who has been spending many of his nights gazing up at her window from the street. They kiss passionately and she suggests they ride around in a cab all night since they have been interrupted by several police officers.
In Act five, it is unclear in the film version what Eliza will do at the play's end. There is a hint that she will stay and run Higgins' household affairs and even that the two of them may become romantically involved.
In the original play, Eliza claims she will marry Freddy, but Higgins seems to think she will stay with him.