In Jane Austen's Mansfield Park, how does Henry's character develop or remain the same throughout the novel? What is the relationship between Mary and Henry Crawford like?

Henry Crawford's character remains the same throughout the novel, though there is a time when he seems on the road to reforming himself. Henry is a bored, wealthy, worldly young man who comes with his sister Mary to stay with their half-sister at the Mansfield parsonage. At first dismissed by Julia and Maria Bertram as not being good looking, he is soon able to use his considerable confidence and charisma to charm them, and he enters into an open flirtation with both in order to amuse himself. Their poor relation, cousin Fanny, is especially shocked at the open flirtation between Maria and Henry because Maria is engaged to another man, the wealthy but socially inept Rushworth.
When Maria marries and Julia goes with her on her honeymoon, Henry decides to amuse himself by making the timid Fanny fall in love with him, hoping, as he confides in Mary, to put a hole in her heart. Instead, Fanny's avoidance and obvious dislike of him, along with her beauty and virtue, cause him to fall in love with her. At this point, it seems the callous rake might reform: he proposes marriage to Fanny and, impressed by her virtue, seems to want to do good, at least to her. Fanny, however, at first thinks the proposal is a cruel joke and later refuses to marry him, knowing what he is. Her rejection only inflames his ardor and determination to win her. When she is banished to Portsmouth, he continues to court her.
But the seeming change of character is only the whim of the moment: Henry soon returns to his selfish ways, seducing the married Maria Bertram and running off with her, and embroiling the entire Bertram family in scandal. He is true to form: he can't stand being rejected and, in society, Maria is cold to him. His own ego at the moment is more important to him than his future contentment or what happens to Maria: he is willing to ruin her life.
Henry is best characterized as the consummate shape shifter, the man who can assume any role to perfection, the serpent in the garden. 
Mary and Henry, growing up together in a hard, dysfunctional household full of "rears and vices," a pun on the ranks of admirals, are extremely close, and Mary works as Henry's co-conspirator in seducing Fanny. Both are worldly, well-socialized and charming, and both have so internalized the materialist and hedonistic values of London society that they have lost their moral compasses. Both throw away happiness through lack of moral imagination.
The key thematic point about Henry's character is not that it changes, because it doesn't, although there is a point where he seems to dimly perceive that Fanny's virtue and her brother William's willingness to work hard to advance in the navy represent a way of life that might be more satisfying than his bored existence filled with idle amusements. The main point is that other people's perceptions of him change based on their own needs and situations. Fanny, for example, while banished to crowded quarters with her dysfunctional family in Portsmouth, begins to waver on Henry and think he may not be so bad a character: environment, Austen says, influences how we perceive others.