While large language models (LLMs) excel at factual recall, the real challenge lies in knowledge application. A gap persists between their ability to answer complex questions and their effectiveness in performing tasks that require that knowledge. We investigate this gap using a patent classification problem that requires deep conceptual understanding to distinguish semantically similar but objectively different patents written in dense, strategic technical language. We find that LLMs often struggle with this distinction. To diagnose the source of these failures, we introduce a framework that decomposes model errors into two categories: missing knowledge and unused knowledge. Our method prompts models to generate clarifying questions and compares three settings -- raw performance, self-answered questions that activate internal knowledge, and externally provided answers that supply missing knowledge (if any). We show that most errors stem from failures to deploy existing knowledge rather than from true knowledge gaps. We also examine how models differ in constructing task-specific question-answer databases. Smaller models tend to generate simpler questions that they, and other models, can retrieve and use effectively, whereas larger models produce more complex questions that are less effective, suggesting complementary strengths across model scales. Together, our findings highlight that shifting evaluation from static fact recall to dynamic knowledge application offers a more informative view of model capabilities.