Large language models (LLMs) are increasingly used to generate summaries from clinical notes. However, their ability to preserve essential diagnostic information remains underexplored, which could lead to serious risks for patient care. This study introduces DistillNote, an evaluation framework for LLM summaries that targets their functional utility by applying the generated summary downstream in a complex clinical prediction task, explicitly quantifying how much prediction signal is retained. We generated over 192,000 LLM summaries from MIMIC-IV clinical notes with increasing compression rates: standard, section-wise, and distilled section-wise. Heart failure diagnosis was chosen as the prediction task, as it requires integrating a wide range of clinical signals. LLMs were fine-tuned on both the original notes and their summaries, and their diagnostic performance was compared using the AUROC metric. We contrasted DistillNote's results with evaluations from LLM-as-judge and clinicians, assessing consistency across different evaluation methods. Summaries generated by LLMs maintained a strong level of heart failure diagnostic signal despite substantial compression. Models trained on the most condensed summaries (about 20 times smaller) achieved an AUROC of 0.92, compared to 0.94 with the original note baseline (97 percent retention). Functional evaluation provided a new lens for medical summary assessment, emphasizing clinical utility as a key dimension of quality. DistillNote introduces a new scalable, task-based method for assessing the functional utility of LLM-generated clinical summaries. Our results detail compression-to-performance tradeoffs from LLM clinical summarization for the first time. The framework is designed to be adaptable to other prediction tasks and clinical domains, aiding data-driven decisions about deploying LLM summarizers in real-world healthcare settings.