As an addition to Tyler's answer, while the story, the locations, the characters and their particular relationships are different from the game, there are still many general similarities in the story development of both the movie and the video game, apart from the obvious presence of the "Sand of Time", the dagger and the "Prince of Persia" himself:
Similar to the movie the Vezier (though not being related to the Prince's family) more or less persuades the Prince's father to conquer a realm for more or less normal reasons (game: wealth and honor, movie: preemptive attack) just to secretly get his hands on the Sand of Time. The Prince in turn tries to impress his father, and steals the Dagger of Time during this attack (not knowing what it can do), much to the dismay of the Vezier's evil plans.
On his way to correct things (though in the game everybody is dead/evil and in the movie he is merely banned/wanted for betrayal), he meets a girl who in the end knows much more about the Sand and the Dagger. She wants to get her hands on it in order to correct things, while the Prince is rather stubborn, doesn't trust her and wants to do things his own way by keeping the dagger getting both into conflict.
At the end, they manage to find the source of the Sand of Time, and by reuniting it with the Dagger, turn back the time to the start of the adventure. The Prince gets the chance to prevent all those evil things happening, and just hints at the girl (who has no memory of her adventure in the other timeline) as to what actually happened.
So while the story is quite different, major plot elements along with the general personalities and relationships of the major characters are not that dissimilar between the movie and the game.