The biggest difference between the play and the 2000 film of Hamlet is the absence of Fortinbras at the end. This, however, is a common elimination in a long play like Hamlet, especially in a production (or film) setting the play in a modern world.
More subtle differences include the elimination of text, especially text that describes events: "Look to the Queen!"; "They bleed on both sides."; "She swoons to see them bleed."; etc. All of these lines are supplanted by visual images. In a performance of the play, where an audience must choose where to look, these lines could be considered necessary, so that each event can attract the audience's attention. In a film, the camera chooses for the audience where to look, so these lines become redundant.
Other alterations to action that is suggested by the text, has to do with the deaths of Laertes, Hamlet and Claudius. In the play, Hamlet is wounded by the poisoned tip of Laertes sword. He then manages to get that sword and wound Laertes in return. Claudius' final demise in the play comes when Hamlet makes him drink the poison that killed Gertrude ("Drink off this potion....Follow my mother."). In the film, in keeping with the modern setting, Laertes shoots Hamlet, who causes Laertes to fire a second shot that mortally wounds him also, and then Hamlet shoots Claudius.
The director adds a visual montage to represent Hamlet's life passing before his eyes as he dies, as well.
It should be noted that these changes are made possible in the film without adding any extra text. The text is still 100% Shakespeare's words, only some of the non-useful text has been removed. Editing of Hamlet, since it's total unedited length can be 4+ hours, is, in fact quite common.