Both authors describe the changing dynamics of the world order in their books. The main difference between them however is about what the future entails for America.
Zakaria argues that the US's global power is declining. Other countries, especially Asian countries like China, are growing and becoming more powerful. Zakaria argues that the US shouldn't resist this change but rather accommodate and cooperate with other countries—this is in the US's own interest.
Nye on the other hand believes that the US still holds certain advantages and will not lose as much power as Zakaria and others suggest. Even though other countries are also moving ahead and influencing the world economy and politics more and more, the US holds unique advantages that others don't have. Nye believes that the US will continue to remain powerful in this new world.