In this paper, the differentially private synchronization control (DPSC) problem is investigated for a class of nonlinear complex networks against possible eavesdropping attacks. The complex network is composed of a number of coupled nodes whose nonlinear dynamics is remotely controlled via a communication link. The initial state of the underly network serve as the private information to be protected from potential eavesdroppers. First, the concepts of mean-square synchronization and differential privacy are introduced to quantify, respectively, the performance of synchronization and the level of privacy. A new DPSC scheme is then proposed, which consists of a noise generator, a pair of input and output noise injectors, and a synchronization controller, in order to guarantee the ultimate mean-square synchronization with preserved differential privacy. In addition, the underlying co-design issue is thoroughly discussed on the input/output noise injector and the synchronization controller. Finally, a series of simulations are carried out to show the effectiveness of the proposed DPSC scheme.