With growing consumer demand for plant-based products that mimic the eating experience of animal-based products, there is a need for improvement in instrumental measurements of sensory texture. This study aimed to characterize textural differences between dairy and non-dairy cheeses, and to investigate whether Large Amplitude Oscillatory Shear (LAOS) rheometry could discriminate sensory texture better than Texture profile analysis. Commercial dairy and non-dairy cheddar, mozzarella, and cream cheese were selected to provide a wide range of textures. Sensory evaluation used the check-all-that-apply methodology with 73 consumers. Texture profile analysis force-distance data were analyzed empirically, and also converted to stress and strain (see https://shiny.csiro.au/texture_dash). The major textural differences between dairy and non-dairy cheddar were related to structural cohesion, according to both instrumental measures (dairy cheddar had 1.5-fold higher failure stress and 2.2-fold higher failure strain) and sensory measurements (dairy cheddar was more chewy and less crumbly). In contrast, cream cheeses showed similar textural properties using sensory testing but significant instrumental differences (non-dairy cream cheese had 5.7-fold higher modulus of deformability, 4.7-fold higher failure stress). For mozzarella, there were large differences in both sensory attributes (chewy, crumbly, jelly-like, stretchy) and instrumental parameters (13.6-fold difference in modulus, 2.7-fold difference in failure stress). LAOS rheometry gave insights into the mechanisms by which samples absorbed or dissipated mechanical energy at nonlinear strains. The LAOS parameter correlated well with sensory attributes creamy, fatty/oily, and moist, indicating the potential of this technique to measure structural phenomena linked to sensory attributes that resonate with consumers.