What distinguishes Montag's approach to life from those of most of the other characters in Fahrenheit 451?

Montag is different because over the course of the book, he evolves from being more or less a mindless drone like everyone else to being introspective and evaluating his job and his life. 
When the story starts, Montag seems to be much like everyone else in his society.  He is a fireman, and he doesn’t question his job or his life.  He is enjoying being a fireman and likes burning books.

IT was a special pleasure to see things eaten, to see things blackened and changed. With the brass nozzle in his fists, with this great python spitting its venomous kerosene upon the world, the blood pounded in his head, and his hands were the hands of some amazing conductor playing all the symphonies of blazing and burning to bring down the tatters and charcoal ruins of history. (Part I)

Mindlessness is part of this existence.  It doesn’t occur to Montag to be introspective until he meets his teenage neighbor, Clarisse.  She asks him if he is happy, and he never thought to ask himself this, so the question shakes him.

"Are you happy?" she said.
"Am I what?" he cried.
But she was gone-running in the moonlight. Her front door shut gently.
"Happy! Of all the nonsense."
He stopped laughing. (Part I)

However, Montag is not like everyone else in that he can change.  The fact that Montag would even think about this question shows that he approaches life differently.  Clarisse is unique, but obviously Montag is too.  He is not like the other members of his community, who never question, never think, and never feel.  They are empty shells.  The fact that books have been banned is only the tip of the iceberg.  People spend all of their time either watching television or listening to their seashell radios in a semi-comatose state.  They drive too fast and never hold real conversations.  
Montag begins to question his job, which is unusual as well.

"I've tried to imagine," said Montag, "just how it would feel. I mean to have firemen burn our houses and our books."
"We haven't any books."
"But if we did have some."
"You got some?"
Beatty blinked slowly.
"No." (Part I)

Reflection and empathy are not common for firemen, or anyone from Montag's society.  The fact that he begins to evaluate his job after Clarisse gets him to question his life really also puts him at risk, however.  It is illegal to have books, and Beatty begins to be suspicious of him.
Another way that Montag is different is the fact that he is curious about books.  Most of the others in his society do not even care what is in the books.  Montag steals books.  Beatty knows what is in the books and claims that he thinks they do not matter.  He teases Montag with them.  Montag, on the other hand, wants more.  He wants to really understand what is in the books.
After stealing the books, Montag wants to know how to read them.  This is why he seeks out Faber.  Mildred, his vacuous wife, has had enough.  She turns him in.  Beatty gets his revenge, but Montag has the last laugh, so to speak.  He turns his flamethrower on Beatty.  Montag never wanted to kill a man, but he had no choice.  Montag ran off and found the book people and a new society.