This paper addresses the challenge of fitting classical material models to experimental data under both incompressible [1] and slightly compressible conditions. In classical, e.g., for elastomers, incompressibility simplifies the formulation, assuming constant volume during deformation, but real-world materials often exhibit slight compressibility, requiring more sophisticated approaches to account for volumetric changes. We discuss the fitting process for compressible material models, compare different volumetric strain energy terms, and vary the bulk modulus to improve the accuracy in capturing the material's response. This study also explores the trade-offs between perfectly incompressible and compressible models in terms of fitting accuracy and computational complexity. In addition to classical approaches, we also exploit physics-augmented neural networks (PANNs) [2] as a flexible alternative for constitutive modeling. These neural networks incorporate physical constraints directly into the network architecture and the training process. Drawing from classical material models, we present a new method to enforce (near) incompressibility within PANNs. We investigate benchmark tests on various datasets to compare the performance of PANNs with classical models, highlighting the accuracy, generalization, and critically discuss the material response to unseen deformation states for both methods. This study also discusses the constraints imposed by different material parameters, offering insights into the strengths and limitations of both classical and neural network-based modeling approaches for capturing the material’s behavior under various load scenarios.