Preheat oven to 325°F. Lightly butter 9-inch-diameter springform pan with 2 3/4-inch-high sides. Stir cracker crumbs and sugar in medium bowl to blend. Add melted butter and stir until evenly moistened. Press crumb mixture firmly onto bottom (not sides) of prepared pan. Bake until crust is set, about 12 minutes. Cool completely. Maintain oven temperature.
Puree mangoes in processor until smooth. Set aside 2 cups mango puree (reserve any remaining puree for another use). Beat cream cheese, sugar, and vanilla in large bowl until smooth. Add eggs 1 at a time, beating well after each addition. Add 2 cups mango puree and beat until well blended. Pour filling over crust in pan.
Bake cake until set and puffed and golden around edges (center may move very slightly when pan is gently shaken), about 1 hour 25 minutes. Cool cake 1 hour. Refrigerate uncovered overnight. Run small knife between cake and sides of pan to loosen. Remove pan sides. Transfer cake to platter. Cut into wedges and serve with sliced mangoes.