Crosslinked polymers are three-dimensional network structures formed by interactions that range from weak physical associations to strong covalent bonding. The inherent complexity in their structures results in rich and diverse rheological response at large deformations. In the present work, we investigate two crosslinked materials based on poly(vinyl alcohol) (PVA) using large amplitude oscillatory shear (LAOS). Physically crosslinked PVA-borax exhibited Giesekus-like linear response, and a gradual intra-cycle strain stiffening and shear thinning at large strains. These signatures are attributed to temporary junction points and stretching of chains between crosslinks in the system. In contrast, chemically crosslinked PVA-hyaluronic acid (PVA-HA) gels exhibited a weak frequency dependence of linear moduli and intra-cycle strain stiffening and shear thickening at large strains. PVA-HA contains permanent crosslinks that move apart, along with stretching of crosslinked and uncrosslinked chains, and these features are responsible for such LAOS signatures. Based on these insights, we propose modifications to existing models and obtain quantitative description of LAOS response of both PVA-borax and PVA-HA crosslinked systems.