I am going to assume you are talking about the more recent film produced in 1996. There are many differences between this film and the play. One of the most notable differences is the death scene near the end of the play. In the play, Romeo dies beside Juliet, the priest enters the tomb, and Juliet awakes to find him cold. She talks with the priest but refuses to leave Romeo's side. After the priest leaves, she stabs herself with a dagger. In the movie, Juliet awakes to find Romeo dieing. He dies in her arms while she looks on in horror. Romeo saw that Juliet was alive before he died, but it was too late to save his life. The priest never comes to the tomb in the movie and Juliet is left alone with Romeo in her arms. She kills herself with his gun and not his dagger. This is another major difference between the play and movie. This particular movie attempted to modernize the story. They used guns with the word dagger or sword etched on the side. They also substituted several lines scripted for a messenger with a TV new anchor. For instance, the news anchor told them about the gala at the Capulet's rather than a servant who could not read. There are many other discrepancies between the play and the movie but these are probably the main two.
I'll interpret your question as the 1996 version of the film.
The difference that is most striking to me is that the movie plays upon gender roles - Mercutio is a transgender character, and thus subverts our expectation from Shakespeare's play that he is masculine. The whole mis-en-scene of the film as a LGBT feel to it; the director Baz Luhrmann is a big supporter of an event called the 'Sydney Gay and Lesbian Mardi Gra' in Australia - and the bright colours and costumes reflect this. In the play, there really is no question about whether men are really 'men' - they joust and fight, and do very masculine things.
Another striking difference is that the movie is set in a place which resembles Miami (in the movie known as 'Verona Beach'). There is nothing that resembles Renaissance Italty (or England) about this place - it is very much 20th century American as a multicultural and multilingual society. Luhrmann believes that the American accent is attuned to Shakepeare's prose, as he notes: "When Shakespeare wrote these plays, they were written for an accent that was much more like an American sound, and when you do Shakespeare with an American accent it makes the language very strong, very alive”