In viewing the film version of Beowulf, how did the story adapt and change? Was that a cultural adaptation?

Since you do not specify which version you mean, I will assume you're talking about the most recent movie adaptation from 2007, which changes quite a bit about the story.
The film adds a dimension of grey morality to the story. This could be perceived as a culture-based adaptation change, since modern society is less comfortable with a belief in pure good versus pure evil. For instance, Beowulf and Hrothgar are presented as good warrior-kings in the poem, while in the film, both men are presented as flawed and power-hungry, seduced by Grendel's mother and impregnating her with beasts that threaten their people (Grendel is now Hrothgar's illegitimate progeny; the Dragon is Beowulf's). At the end of the movie, the heroic Wiglaf is implied to fall prey to the temptation of Grendel's mother as well, subverting not only his noble characterization in the original poem, but the poem's ideas about heroism in general.
Film Grendel is less a vicious, antisocial beast than an outcast wronged by society, which is a more postmodern way of looking at the character. Contemporary fiction and film often recast traditional monsters like vampires or werewolves as misunderstood victims of social intolerance, so that could be seen as a cultural change as well. By complicating the motivations of the characters, the filmmakers seek to retell the epic in a way which they feel will connect with a modern audience less likely to buy into stark moralistic duality.
The tension between paganism and Christianity is also multiplied in the film. In the poem, the poet narrator is a Christian telling a story of pagan heroism. As a result, there is a contrast between Christian mercy and faith in an all-loving god, and the individualistic heroism of the pagan warrior-kings. The film brings this understated idea explicitly into the narrative, making Unferth a Christian and showing the influence of Christianity gradually overtaking Beowulf's pagan society.