Although auditory cortex is thought to play an important role in processing complex natural sounds such as speech and animal vocalizations, the specific functional roles of cortical receptive fields (RFs) remain unclear. Here, we study the relationship between a behaviorally important function: the discrimination of natural sounds and the structure of cortical RFs. We examine this problem in the model system of songbirds, using a computational approach. First, we constructed model neurons based on the spectral temporal RF (STRF), a widely used description of auditory cortical RFs. We focused on delayed inhibitory STRFs, a class of STRFs experimentally observed in primary auditory cortex (ACx) and its analog in songbirds (field L), which consist of an excitatory subregion and a delayed inhibitory subregion cotuned to a characteristic frequency. We quantified the discrimination of birdsongs by model neurons, examining both the dynamics and temporal resolution of discrimination, using a recently proposed spike distance metric (SDM). We found that single model neurons with delayed inhibitory STRFs can discriminate accurately between songs. Discrimination improves dramatically when the temporal structure of the neural response at fine timescales is considered. When we compared discrimination by model neurons with and without the inhibitory subregion, we found that the presence of the inhibitory subregion can improve discrimination. Finally, we modeled a cortical microcircuit with delayed synaptic inhibition, a candidate mechanism underlying delayed inhibitory STRFs, and showed that blocking inhibition in this model circuit degrades discrimination.