1. Make cake: Preheat oven to 325°F. Butter an 8-inch square, nonstick pan.
2. In a saucepan, melt butter with sugar over medium heat, stirring until butter has melted and sugar has dissolved. Remove from heat; whisk in applesauce. In a large bowl, whisk flour, baking powder, baking soda, cinnamon and salt. Add applesauce mixture and stir until just combined.
3. Pour batter into prepared pan. Bake until a toothpick inserted into center comes out clean, 35 to 40 minutes. Let cool in pan on a wire rack for 10 minutes, then turn out onto rack to cool completely.
4. Make frosting: Using an electric mixer on medium-high speed, beat cream cheese and butter until blended. Beat in confectioners' sugar and maple syrup until smooth. Spread over cake; serve.