The recurring theme in The Matrix trilogy is a single word: choice.
Neo does not "believe" he is The One, and does not exert himself as if he can remake the Matrix as he sees fit, per Morpheus's description of what the defining trait of The One is. It is only when he is faced with the fork of pulling the plug on Morpheus or attempting to save him does he start to realize it himself.
"The Oracle... she told me this would happen, she told me... that I
would have to make a choice." - Neo
Later Morpheus reinforces this idea, before Neo can express his self-doubt:
"Neo, sooner or later you're going to realize just as I did that
there's a difference between knowing the path and walking the path." - Morpheus, to Neo
Finally, when faced with what should be the no-brainer decision of running from Agent Smith, he stops. He stops because he can; he does not believe himself to be The One, he does not question that this Agent can physically best him, but he stops because he can. In the movie's final minutes, Neo stops the bullets from the guns of the united Agents with a single word, defends himself from Agent Smith's attempted attack with only one arm and rejects him with a single kick. We cannot know what he is experiencing at this point, but he appears to simply be choosing to deny what is happening around him. Bullets fly; he rejects their premise of inertia (it doesn't exist anyway, at least not in the Matrix). Smith attacks; he spends no effort in retaliating, his expression is focused and calm. He simply chooses not to be hit.
Later in the series we see that Smith is alive and well, and has a compulsion for disobedience. He insists this is the result of copied or overwritten data:
Smith: I don't fully understand how it happened, perhaps some part of
you imprinted on to me, something overwritten or copied, it is at this
point irrelevant. What matters is whatever happened, happened for a
Neo: And what reason is that?
Smith: I killed you Mr. Anderson, I watched you die, with a certain
satisfaction I might add. And then something happened. Something that
I knew was impossible but it happened anyway. You destroyed me Mr.
Anderson. Afterward, I knew the rules, I understood what I was
supposed to do, but I didn't. I couldn't. I was compelled to stay,
compelled to disobey. And now here I stand because of you Mr.
Anderson, because of you, I am no longer an Agent of this system,
because of you I've changed, I'm unplugged, a new man so to speak.
Like you, apparently, free.
Smith is burdened with an overabundance of choice; where he once fulfilled his duties and obeyed the cause and effect of the world he existed in, he now sees that he need not do any of that. Unfortunately he perceives this as being robbed of purpose, and goes on to screw up all the things.
As the movie continues, we are given crucial insight into the secret and stumbling block of The Matrix; it is based on a flawed premise in order to facilitate the unanticipated behavior of conscious humans which reject the mental prison of The Matrix.
The Architect: As I was saying, she stumbled upon a solution whereby
nearly ninety-nine percent of the test subjects accepted the program
provided they were given a choice - even if they were only aware of it
at a near-unconscious level. While this solution worked, it was
fundamentally flawed, creating the otherwise contradictory systemic
anomaly, that, if left unchecked, might threaten the system itself.
Ergo, those who refused the program, while a minority, would
constitute an escalating probability of disaster.
Indeed - even with Smith's considerable ability to match Neo point-for-point in the last movie, he is unable to force his submission and when prompted to answer:
Smith: "Why, Mr. Anderson? Why do you do it? Why get up? Why keep
fighting? Do you believe you're fighting for something? For more than
your survival? Can you tell me what it is? Do you even know? Is it
freedom? Or truth? Perhaps peace? Yes? No? Could it be for love?
Illusions, Mr. Anderson. Vagaries of perception. The temporary
constructs of a feeble human intellect trying desperately to justify
an existence that is without meaning or purpose. And all of them as
artificial as the Matrix itself, although only a human mind could
invent something as insipid as love. You must be able to see it, Mr.
Anderson. You must know it by now. You can't win. It's pointless to
keep fighting. Why, Mr. Anderson? Why? Why do you persist?"
Neo: "Because I choose to.
Virtually every aspect of the major themes of The Matrix revolve around choice's role in existence. Whereas it is empowering to humans such as Neo, Morpheus, and Trinity, the antagonist Smith has to connect it to a reason. It must be a means to an end, not an end unto itself. It causes fluctuations in the code of the Matrix which require the clean install of the prime program every 100 years or so. The Matrix trilogy uses the ramifications of 'the ability to choose' as the single-most defining difference between human and machine consciousness.