Graph autoencoders (GAEs), a class of generative self-supervised learning methods, have demonstrated great potential in recent years. Typically, GAEs employ an encoder to map the input graph into a latent representation and a decoder to reconstruct the graph by recovering its characteristics, such as node features or structural information. However, GAEs that rely on feature reconstruction often fail to recover the unique information that differs from neighboring nodes, leading to excessive feature smoothness between neighboring nodes and sub-optimal performance. To address this issue, we propose two complementary strategies applied during the encoding and decoding phases, respectively. At the decoding stage, we develop a simple yet effective approach to preserve the distinctiveness between neighbors in the raw graph. We conceptualize the encoder-decoder architecture of GAEs as a teacher-student framework, where we compute pairwise node dissimilarities in both the original and reconstructed graphs and enforce a Kullback-Leibler divergence constraint to transfer distinctiveness from the input to the output space. At the encoding stage, we introduce a discriminative constraint that encourages decorrelation among similar node pairs, implemented via a covariance-based regularization that jointly considers node and neighborhood embeddings. Based on these strategies, we present ClearGAE, a GAE capable of reconstructing graphs while preserving their essential distinctions. Extensive experiments on three types of graph tasks demonstrate the effectiveness of ClearGAE. Moreover, our strategies are model-agnostic and can be seamlessly integrated as plug-and-play modules into other GAE variants.