The latest advances in full-reference image quality assessment (IQA) involve unifying structure and texture similarity based on deep representations. The resulting Deep Image Structure and Texture Similarity (DISTS) metric, however, makes rather global quality measurements, ignoring the fact that natural photographic images are locally structured and textured across space and scale. In this paper, we describe a locally adaptive structure and texture similarity index for full-reference IQA, which we term A-DISTS. Specifically, we rely on a single statistical feature, namely the dispersion index, to localize texture regions at different scales. The estimated probability (of one patch being texture) is in turn used to adaptively pool local structure and texture measurements. The resulting A-DISTS is adapted to local image content, and is free of expensive human perceptual scores for supervised training. We demonstrate the advantages of A-DISTS in terms of correlation with human data on ten IQA databases and optimization of single image super-resolution methods.