The world in which we are living is a huge network of networks and should be described by interdependent networks. The interdependence between networks significantly affects the evolutionary dynamics of cooperation on them. Meanwhile, due to the diversity and complexity of social and biological systems, players on different networks may not interact with each other by the same way, which should be described by multiple models in evolutionary game theory, such as the Prisoner's Dilemma and Snowdrift Game. We therefore study the evolutionary dynamics of cooperation on two interdependent networks playing different games respectively. We clearly evidence that, with the increment of network interdependence, the evolution of cooperation is dramatically promoted on the network playing Prisoner's Dilemma. The cooperation level of the network playing Snowdrift Game reduces correspondingly, although it is almost invisible. In particular, there exists an optimal intermediate region of network interdependence maximizing the growth rate of the evolution of cooperation on the network playing Prisoner's Dilemma. Remarkably, players contacting with other network have advantage in the evolution of cooperation than the others on the same network.