Large Language Models (LLMs) have shown the potential to significantly contribute to patient care, diagnostics, and administrative processes. Emerging biomedical LLMs address healthcare-specific challenges, including privacy demands and computational constraints. However, evaluation of these models has primarily been limited to non-clinical tasks, which do not reflect the complexity of practical clinical applications. Additionally, there has been no thorough comparison between biomedical and general-domain LLMs for clinical tasks. To fill this gap, we present the Clinical Language Understanding Evaluation (CLUE), a benchmark tailored to evaluate LLMs on real-world clinical tasks. CLUE includes two novel datasets derived from MIMIC IV discharge letters and four existing tasks designed to test the practical applicability of LLMs in healthcare settings. Our evaluation covers several biomedical and general domain LLMs, providing insights into their clinical performance and applicability. CLUE represents a step towards a standardized approach to evaluating and developing LLMs in healthcare to align future model development with the real-world needs of clinical application. We publish our evaluation and data generation scripts: https://github.com/dadaamin/CLUE