In the film, after Katniss shoots and kills President Coin, the crowd of rebels behind her rushes forward and attacks Snow-I would presume that Snow was lynched by the angry mob. However, in the book, it is debated whether he died from choking on his blood or being trampled to death. Either way, the cruel, heartless dictator died an inglorious death-it didn't really matter anyway how he died. I think Collins allowed President Snow to die in such a matter to symbolize that Snow's death didn't really matter anymore. Sure, he was once the supreme ruler of Panem, but by the end of Mockingjay, he is nothing but just, well, an elderly man. Whether he died or not, the point being, Snow was already symbolically dead, in that he no longer had real power or influence in Panem, Coin was calling the shots! Snow was simply what remained of the old Panem, which was already overthrown/destroyed. At that point, it didn't really matter whether President Snow lived or died, his regime was toppled and his time was over. This is what Collins desired to show, that Snow didn't really matter at this point.
By contrast, there were several hints from various characters that President Coin had questionable motives. While Squad 451 began to infiltrate the Capitol, Boggs told Katniss that President Coin never liked her, and wanted to rescue Peeta from the arena instead. Of course, the others didn't agree, and Katniss ended up being rescued instead. Boggs asked Katniss the important question: Who will Katniss support for the post-rebellion President of Panem? When Katniss responded that she never thought about it, Boggs warned that if she her decisive answer was Alma Coin, she would be seen as a threat. In other words, from what Boggs knew, Alma Coin saw Katniss as a thorn in her side, and the most Coin and Everdeen did for each other was to tolerate.
Following Katniss' conversation with Snow following the invasion of the Capitol, Paylor asked her "Did you find what you were looking for?" Paylor was also the person who demanded that the soldiers guarding the imprisoned President Snow to allow Katniss to speak with him. In the film, when Coin was proposing the 76th Hunger Games (with Capitol Children) to the remaining victors, Haymitch's attitude was of concern. When Coin mentioned "I have taken the burden and honor of declaring myself interim President of Panem", Haymitch seemed skeptical, "Interim? Exactly-how long is this interim?" He seemed to have his suspicions about President Coin.
Snow told Katniss "Make no mistake, she (Coin) intends to take my place now".
From all these hints, it is safe to assume that President Coin had her own, selfish motives, from the attitudes of several characters. Snow, on the other hand, was already dead, figuratively speaking. He had lost all control over Panem, and the death of Coin was what really changed the course of Panem's post-war history.
Following Coin's assassination, an election was held instantly and Paylor became the new president, and the Hunger Games were permanently ended, and Panem was transformed into the Republic that Plutarch mentioned they would set up, following the war. Compare that with Coin's promise "We'll plan an election when the time is right." I'd have serious reservations about some new political leader who would say THAT!
