He sees himself as
the chosen champion. and hero.
Yet once given that power (and responsibility that came with it) , could never let go of it, would never turn back into his suposedly "weak" de-powered form, or worse never "share" the power with what he saw as lesser beings.
In the end that was his undoing, and why the wizard turned against him.
In looking for a new champion, the wizard found the one kid who given all the great power, would still turn back into a boy, (even one with all his problems in a world that hurt him and robed him of everything he loved).
And the first meaningful thing he does with the power is share it. (that cuts to the core of the character and difference)
How many kids given power would instinctually do that?
This sharing of the power defied all the Wizard's expectations of modern humanity, and revealed why Billy represented the true champion.
Adam has no trust or faith in self-serving, selfish, cruel humanity (who also betrayed him), and sees them as unfit to rule themselves, hence he sees himself as their ultimate ruler above them. He perceives that trust or faith in others, as a flaw or weakness in Billy, and a waste of the power.
There is more to it, and plenty of room for more interpretation, but at least somewhat in it's simplest form, how I understand the main difference.