Large language models (LLMs) are increasingly used to simulate or automate human behavior in complex sequential decision-making settings. A natural question is then whether LLMs exhibit similar decision-making behavior to humans, and can achieve comparable (or superior) performance. In this work, we focus on the exploration-exploitation (E&E) tradeoff, a fundamental aspect of dynamic decision-making under uncertainty. We employ canonical multi-armed bandit (MAB) experiments introduced in the cognitive science and psychiatry literature to conduct a comparative study of the E&E strategies of LLMs, humans, and MAB algorithms. We use interpretable choice models to capture the E&E strategies of the agents and investigate how enabling thinking traces, through both prompting strategies and thinking models, shapes LLM decision-making. We find that enabling thinking in LLMs shifts their behavior toward more human-like behavior, characterized by a mix of random and directed exploration. In a simple stationary setting, thinking-enabled LLMs exhibit similar levels of random and directed exploration compared to humans. However, in more complex, non-stationary environments, LLMs struggle to match human adaptability, particularly in effective directed exploration, despite achieving similar regret in certain scenarios. Our findings highlight both the promise and limits of LLMs as simulators of human behavior and tools for automated decision-making and point to potential areas for improvement.