How do the events shown in the movie 2081 compare to the events in the short story "Harrison Bergeron"?

Great question! The film 2081 closely follows the plot and dialogue of Kurt Vonnegut Jr’s short story “Harrison Bergeron,” but there are several differences.
The opening lines of the film and story are identical. In both, they are used to introduce the dystopian society, the rationale of handicapping for equality and the Handicapper General. However, in the film version, much of the dialogue between characters is condensed to be grittier and more realistic. Some of Vonnegut’s lines are flowery, such as Harrison claiming to be "Emperor of the world." Most similar lines are cut from the film 2081.
Harrison’s handicaps are also less visible. For example, he does not have the “red nose” from Vonnegut’s story, and several other handicaps are missing. Little details from the story, like the exact handicaps or description of Harrison, are modified throughout the film. These minor artistic changes were likely made to render the cinematic portrayal of the story more palatable for a modern audience.
The primary difference between 2081 and “Harrison Bergeron” is the extent of Harrison’s planning. In Vonnegut’s short story, Harrison’s storming of the dance stage and dancing with the ballerina appears to be an unplanned act of defiance after his escape from prison. In 2081, Harrison is presented as a freedom fighter with a carefully premeditated plan. For instance, Harrison orchestrates the “bomb-switch” to re-activate the live broadcast, defeating the technologically advanced censorship. This is no easy feat! Harrison's death on camera indicates a carefully calculated martyrdom with some sort of purpose motivating it.
This major change in the film’s ending impacts the interpretation of the story. Vonnegut’s tale ends with hopelessness. Harrison’s father is too distracted by his intellectual handicap to even remember why he is crying after witnessing the death of his son! In the film 2081, although the father’s response remains the same, the deliberate nature of Harrison’s act seems to have implications beyond his death. It is possible that 2081 hints that Harrison’s death will spark some sort of revolution against the dystopian system.
I hope this helps!