The difference here, between how Snow realizes it and how Finnick realizes it are two completely different things.
Snow is convinced, and he knows that the best way to get to Katniss is not through her family, it’s not through Gale, or her District…it’s through Peeta. He knows now that if he can get to Peeta, he can get to Katniss. Snow sees an opportunity, and as soon as the arena starts to fall apart he wastes no time retrieving Peeta, because he wants to break Katniss. He wants her to suffer, and he knows the best way to do that is by hurting Peeta.
The way Finnick sees it, is that he had misjudged her. He’s been wrong. They all had. Haymitch told them to make sure Peeta stays alive or otherwise Katniss would give up. Haymitch is smart, he knows she loves him before this, “you could live a hundred lifetimes and not deserve him, you know." Finnick is seeing here, that Katniss would rather die than let Peeta die, so he brings him back to life, and as soon as he does, he knows they had her pegged all wrong. And he feels guilty for a good portion of Mockingkay too, because he didn’t warn her, because he thought it was all just an act, the loving Peeta part, and he states that they all just assumed she’d carry on with that.
"But it wasn’t until Peeta hit the force field and nearly died that I—that I knew I misjudged you. That you do love him. I’m not saying in what way. But anyone paying attention could see how much you care about him.”
She convinced Snow. He finally had something to take from her, he tries unsuccessfully to take the entire district, but Peeta…he gets Peeta. He hijacks Peeta.
And, get this…Finnick blames himself, for not telling her about the plan…for not warning her…because now there’s no doubt in his mind that Katniss loves Peeta, and he becomes as afraid for her as he is for himself. For Peeta and Annie in the Capitol, and the two of them left to suffer together.
Anyone? On Snow’s visit before the Victory Tour, he challenged me to erase any doubts of my love for Peeta. “Convince me,” Snow said. It seems, under that hot pink sky with Peeta’s life in limbo, I finally did. And in doing so, I gave him the weapon he needed to break me.
And she did…she convinced them all…
people are looking to you, Katniss. You’ve given them an opportunity. They just have to be brave enough to take it.
“No one really needs me,” he says, and there’s no self pity in his voice. It’s true his family doesn’t need him. They will mourn him, as will a handful of friends. But they will get on. Even Haymitch, with the help of a lot of white liquor, will get on. I realize only one person will be damaged beyond repair if Peeta die