It isn't uncommon for a rival to use the middle and lower class to upstage the incumbent, but the middle and lower class are often back where they were after the coup.
Snow gave Katniss a few key pieces of information, and reminded her of other pieces she knew but may not have considered:
- Coin isn't much different than snow. Both bad, though perhaps different methods of bad.
- Snow was dead already - politically and soon physically. His death in front of everyone is just more gamesmanship.
- She won, but she was handing the leadership over to someone just as bad.
- There was a small window to rebel against the new leader which wouldn't require all the effort that she went through to rebel against the old leader. In fact, Coin undoubtedly knew that after Snow, Katniss was her greatest liability/enemy, and Katniss had already shown significant rebellion in Coin's organization.
- Coin killed Prim. Coin knew that Katniss would eventually learn this, making Katniss that much more dangerous.
Coin manipulated Katniss into becoming the face of the revolution, but at the last moment Snow was able to manipulate Katniss into taking out Coin.
In all this, Katniss is only provided the illusion of free agency, but at every turn her decisions were guided and led down a very narrow path and she really had no choice. Her sister was killed by Coin, and the only choice she really had was to let the old leader die, and get rid of the new leader. By doing so, she allowed the country to have a real choice - something she has been denied her entire life. Yet it was Snow that manipulated her into this path of revenge. Again, she had no choice.
So Snow laughed. He laughed at Coin. He laughed at Katniss. He laughed at the easily manipulated people under his thumb.