Sawyer tells Miles the truth, captures Kate after she crashes her car

In the flash sideways, Sawyer calls Miles to his car outside the police station and shows him the "Sawyer" folder. He tells Miles about his parents' deaths when he was 9 and that he's been hunting down "Sawyer" since he graduated from the police academy. He was chasing down a lead in Australia and got the name Anthony Cooper and has been calling them one by one (which we saw him doing earlier in the episode). Sawyer says that he plans to kill "Sawyer" once he finds him and Miles says that if he would have known, he would have tried to talk him out of this hunt.

Suddenly, a blue car slams into their car from behind. A hooded  person jumps out of the car and starts running. Sawyer and Miles chase cut them of at the end of an alley and Sawyer slams the perp against a chain fence. He spins the person around to reveal Kate and Sawyer is stunned to see the woman he helped escape at the airport.

So clearly, Sawyer in the the flash sideways is very, very similar to himself in the original timeline, hence him helping Kate escape at the airport. It really seems that the only major difference here is his occupation and his friendship with Miles. He's still consumed with vengeance, but in a more controlled environment.