Chigurh walks away from the accident, not to be seen again.
In the end, Ed Tom (Jones) has already decided on retirement (life) rather than bringing the killer (Bardem) to justice. He knows he found the killer. The Sheriff retires because he realizes that the country he covers (south Texas) is No Country for Old Men. So in the end, he sits at the table and tells his wife about a dream: he heads into snowy, dangerous mountains (his life as a sheriff), and a man goes on ahead of him to await him (the killer still awaits). And then he woke up to the fact that in real life he was acting out this dream. So in the end, rather than live the nightmarish, lethal continuation of the dream, he wakes up to choosing the safer life of retirement. He has taken the lesson from his forcibly retired colleague in the wheelchair: a life of disability isn't worth it.
Actually, the Coen brothers left a big clue when Sheriff Ed Tom goes to the hotel room (behind the crime scene yellow tape) where the assassin, Chigurh, waits. Chigurh has shot the lock cylinder out, indicating to Ed that it's Chigurh's doing, and BOTH of them watch the other's movement in the reflection in the shiny lock tube. Ed Tom draws his pistol and enters for a search, and realizes the killer is still within, given the locked window in the far room.
"Momma, take this badge offa me, ... I feel I'm knockin on heaven's door." -- Bob Dylan, and it's beautifully captured on Jones' old, wincing face when he sits on the bed. So rather than shoot it out, he trusts the killer not to kill him, and he walks out, without pushing it to the point where the killer's other victims utter, "You don't have to do that."
So rather than risk his life, he lets go of capturing his quarry and retires.
Recall that the Sheriff is narrating the past at the beginning of the movie, a clue that he's still alive after the action of the rest of the movie.
Telling his wife of the dream is the final act of letting go of his job, and the movie screen goes black.