Why did Cassian Andor abandon his mission?

1. He was tired of killing people without being punished
2. He saw Lily's eyes in Harry Jyn's eyes in Galen and that was the last straw, since he cared about Jyn.

Official novelization by Alexander Freed covered this in detail (being a book, it's easier to show what's in someone's head).

Destroying Jyn - that’s what it would be, you can admit that much—was his best option. If she did realize what he’d done, she’d turn that feral need against him. She’d want him dead, probably sway the Guardians of the Whills and Bodhi against him as well.
The engineers were arrayed in front of Krennic and Galen. Krennic’s retinue of black-clad troopers fanned out. A few more steps…
Maybe that wouldn’t be such a terrible way to go. He’d assassinated better men than Galen—an Imperial collaborator, the man who’d built a planet killer, remorse be damned. And if Jyn came after Cassian, he’d die for his crimes. There were worse deaths.
Was that what it had come to?
Galen stepped forward. Cassian had the shot.
But he was breathing too hard now. The rifle rose and fell. He clamped a hand on the barrel, lodged it firmly against the rocks.
He was tired of crimes he never answered for.
The Death Star is your answer. Finish this mission, and all is forgiven.
He looked at Galen Erso through his scope and saw his daughter’s eyes.
With a hoarse and ragged cry, he swept the rifle away from the rocks and set it in the mud at his side.