It's not simply a matter of the astonishing growth spurts that he and his friends have experienced since the release of the last movie in 2007 or the fact that, in this film, more than in any other, Harry and his friends are navigating the minefield of adolescent love and lust. Instead, most reviews find that Harry's transition to stem from his pursuit of Voldemort's past with Dumbledore, marking him as "no longer the Boy Wonder at all but Dumbledore's right hand man."
I can't fault reviewers for mistakenly identifying the struggles above as the marker for Harry's transition out of adolescence. After all, Harry's most heroic moment has been cut from the film in favor of a manufactured confrontation with two Death Eaters (followers of Voldemort) over the Christmas holidays. This scene replaces a telling confrontation between Harry and the new Minister of Magic, Rufus Scrimgeour.
Scrimgeour's subplot, which has been excised entirely from the film, pushed Half Blood Prince beyond a simple, Manichean struggle against external forces of darkness. In the book, J.K. Rowling did an admirable job balancing the threat of Voldemort's unmitigated evil with the moral transgressions of Scrimgeour and the rest of Harry's government.
In the book, Scrimgeour, in an effort to appear resolute in his 'War on (Magical) Terror,' has locked up any number of innocent bystanders, including Stan Shunpike, a nebbishy bit player seen only briefly in books three and four. Stan has been thrown into Azkaban, the wizarding world's most degrading and hellish prison on the mere suspicion of consorting with Voldemort's followers.
Rather than admit an error, and desperate to appear to be making progress against a terrifying enemy, Scrimgeour refuses to release Shunpike. When the book was released in 2005, it invited comparisons to George W. Bush's horrific treatment of detainees in Guantanamo.
Like Bush, Scrimgeour is described as a "decisive... man of action," and, just like Bush, he knows the value of a good photo op. To raise the morale of the wizarding world, Scrimgeour comes to find Harry at Christmas to ask him to make a public show of support for the Ministry.
In the book, standing in the front yard of the Weasley home, Harry faces a choice where the stakes seem as low as they could be. Far from fighting an apocalyptic wizarding duel, Harry never so much lays a finger on his wand, and he's standing opposite a man who opposes Voldemort every bit as fiercely as Harry himself.
Nevertheless, this confrontation is one of the most satisfying in the book, as Harry refuses Scrimgeour's request, asking, "Won't that seem as though I approve of what the Ministry's up to? ...You see, I don't like some of the things the Ministry is doing. Locking up Stan Shunpike for example."
There's no spell that Harry can cast to free Stan Shunpike or to make Scrimgeour change his mind. After one more attempt to persuade Harry, Scrimgeour leaves, and both he and Harry return to the task of defeating Voldemort.
Part of growing up is finding a way to deal with problems that can't be dispelled with the flick of a wand. Growing up requires us to find a way to cope with the death of our mentors and idols. We must face not only the physical death suffered by one of Harry's mentors but the death of our ability to blindly idealize our leaders, to accept them as unquestionably good.
A battle against a self-styled 'Dark Lord' is the battle of a child. It's a battle reserved for a world divided neatly into good people and Death Eaters. Stan Shunpike's imprisonment challenges Harry to confront injustices on a scale less epic but more difficult.
Harry struggles throughout both the book and movie versions of Half Blood Prince to accept the possibility of the goodness of Severus Snape, or, more essentially, that someone can be one of the good guys, while remaining a fundamentally unpleasant person. Scrimgeour presents the opposite problem: a man so committed to opposing evil that he has neglected to be good.
We don't admire fantasy heroes like Harry for their stellar wandwork or potions prowess but for their strength of character and determination to do the right thing. A true hero practices constant vigilance because the world needs to be saved every single day. A true hero remembers that, sometimes, the world that needs saving is closer (and smaller-looking) than big budget epics would teach us to expect.