Graph contrastive learning (GCL) has achieved remarkable success in graph self-supervised learning (SSL) through an augmenting-contrasting paradigm. Existing augmentation strategies typically generate augmentations independently, ignoring the explicit modeling of the underlying relationship between augmentations, i.e., augmentation discrepancy. In addition, previous discrete augmentations (e.g., edge dropping and feature masking) also hinder the path toward joint optimization. These limit the diversity and complementarity of augmentations, leading to the suboptimal contrastive learning. In this article, we propose a novel adversarial augmentation method, called adversarial augmentation with maximum discrepancy for GCL (AMD-GCL), to jointly optimize pairwise augmentations. The core of AMD-GCL is an adversarial augmentation constraint module that maximizes the discrepancy between pairwise augmentations. Specifically, we establish a theoretical analysis indicating that maximizing graph reconstruction error in a continuous space serves as a surrogate for minimizing mutual information (MI), laying the basis for the differentiable constraint of augmentation discrepancy. Based on this, AMD-GCL designs a min-max problem. We directly add continuous adversarial perturbations to the original graph structure and features to maximize the reconstruction error. Meanwhile, we maximize the reconstruction error between pairwise augmentations to amplify the discrepancy. This leads to a maximization problem. After obtaining augmentations, AMD-GCL optimizes both the contrastive loss and reconstruction objectives, deriving a unified minimization problem. The adversarial augmentations are iteratively updated during the training process. Comprehensive experiments on 18 datasets demonstrate the superiority and robustness of AMD-GCL on several downstream tasks and various adversarial scenarios.