Wargames have a long history in the development of military strategy and the response of nations to threats or attacks. The advent of artificial intelligence (AI) promises better decision-making and increased military effectiveness. However, there is still debate about how AI systems, especially large language models (LLMs), behave as compared to humans. To this end, we use a wargame experiment with 107 national security expert human players designed to look at crisis escalation in a fictional US-China scenario and compare human players to LLM-simulated responses. We find considerable agreement in the LLM and human responses but also significant quantitative and qualitative differences between simulated and human players in the wargame, motivating caution to policymakers before handing over autonomy or following AI-based strategy recommendations.