What is the most important element in Branagh's Hamlet that illustrates the uniqueness of his approach to the play?
An important element of the unique approach to Hamlet in Branagh's film version is that he included flashbacks and scenes that don't exist in Shakespeare's play. So, in effect, he created his own story of Hamlet, which can't be said to be Shakespeare's play since we can't attribute these additions to Shakespeare.
Additions he made include a scene between Hamlet and Yorick. Yorick is the name of the man's skull found by the gravedigger, mentioned as the King's jester. Branagh gave Yorick and Hamlet a past relationship through flashback.
He also took the story into his own hands and showed Hamlet and Ophelia in bed together, thereby answering a question that the play leaves unanswered: Has Ophelia given her virginity to Hamlet before the events of the play?
I don't know if the addition of these scenes would be agreed upon by everyone as the most "unique" element in Branagh's approach to the text, but it certainly sets the film apart from other filmed versions of Hamlet and sets a definitive stamp of interpretation on the film.