The use of large language models (LLMs) in hiring promises to streamline candidate screening, but it also raises serious concerns regarding accuracy and algorithmic bias where sufficient safeguards are not in place. In this work, we benchmark several state-of-the-art foundational LLMs - including models from OpenAI, Anthropic, Google, Meta, and Deepseek, and compare them with our proprietary domain-specific hiring model (Match Score) for job candidate matching. We evaluate each model's predictive accuracy (ROC AUC, Precision-Recall AUC, F1-score) and fairness (impact ratio of cut-off analysis across declared gender, race, and intersectional subgroups). Our experiments on a dataset of roughly 10,000 real-world recent candidate-job pairs show that Match Score outperforms the general-purpose LLMs on accuracy (ROC AUC 0.85 vs 0.77) and achieves significantly more equitable outcomes across demographic groups. Notably, Match Score attains a minimum race-wise impact ratio of 0.957 (near-parity), versus 0.809 or lower for the best LLMs, (0.906 vs 0.773 for the intersectionals, respectively). We discuss why pretraining biases may cause LLMs with insufficient safeguards to propagate societal biases in hiring scenarios, whereas a bespoke supervised model can more effectively mitigate these biases. Our findings highlight the importance of domain-specific modeling and bias auditing when deploying AI in high-stakes domains such as hiring, and caution against relying on off-the-shelf LLMs for such tasks without extensive fairness safeguards. Furthermore, we show with empirical evidence that there shouldn't be a dichotomy between choosing accuracy and fairness in hiring: a well-designed algorithm can achieve both accuracy in hiring and fairness in outcomes.