Turpidus spares Case intentionally because that puts her in his debt, and he gets to annoy Angalacon as a bonus (remember, they dislike each other). And that gives me an excuse to write more into their relationship than just "hey, these dudes don't like each other because reasons x, y, and z."
"I'm fighting because there's a battle to be fought, Archer. I'm fighting to win. That's all."
"The answer should be obvious. I choose the side with the highest chance of victory, however slight."