God importance of food in HUNGER Games is such amazing aspect of the books and shows the mentality of both Katniss AND Snow.
Coriolanus learns that food is power. He sees Nero Prize cutting off maid's leg to eat it. He sees what tributes will do for food. He knows what Lucy Gray did for food. He knows what he would do for food.
In his eyes food is a luxury and ultimate means of manipulation. That's why there is a tessare system, that's why there are monthly packages for the winning districts, that's why Games are a yearly public spectacle in the Capitol. He keeps districts hungry for food and Capitol hungry for entertainment.
Katniss learns that food is love. It starts with Peeta throwing her the burned loaf of bread. Then goes further into her love for Prim, which is the main cause of her hunting. Then with Gale as her hunting partner. Then with Madge who is her best friend and loves strawberries. Then with Mr. Mellark who loves squirrels. Then with Rue and bread from District 11. Then with Peeta again, with the berries.
Katniss doesn't use the power of food over people. She shares it. That's how she builds connections, forms friendships, wins over people's hearts, starts and wins revolutions.
For Snow food it a tool with which you can sew starvation and chaos.
For Katniss food is a tool with which you can form bonds and find peace.