Federated Learning (FL) is a collaborative learning framework that enables edge devices to collaboratively learn a global model while keeping raw data locally. Although FL avoids leaking direct information from local datasets, sensitive information can still be inferred from the shared models. To address the privacy issue in FL, differential privacy (DP) mechanisms are leveraged to provide formal privacy guarantee. However, when deploying FL at the wireless edge with over-the-air computation, ensuring client-level DP faces significant challenges. In this paper, we propose a novel wireless FL scheme called private federated edge learning with sparsification (PFELS) to provide client-level DP guarantee with intrinsic channel noise while reducing communication and energy overhead and improving model accuracy. The key idea of PFELS is for each device to first compress its model update and then adaptively design the transmit power of the compressed model update according to the wireless channel status without any artificial noise addition. We provide a privacy analysis for PFELS and prove the convergence of PFELS under general non-convex and non-IID settings. Experimental results show that compared with prior work, PFELS can improve the accuracy with the same DP guarantee and save communication and energy costs simultaneously.