Climate change is a major socio-scientific issue shapes public decision-making and policy discussions. As large language models (LLMs) increasingly serve as an interface for accessing climate knowledge, whether existing benchmarks reflect user needs is critical for evaluating LLM in real-world settings. We propose a Proactive Knowledge Behaviors Framework that captures the different human-human and human-AI knowledge seeking and provision behaviors. We further develop a Topic-Intent-Form taxonomy and apply it to analyze climate-related data representing different knowledge behaviors. Our results reveal a substantial mismatch between current benchmarks and real-world user needs, while knowledge interaction patterns between humans and LLMs closely resemble those in human-human interactions. These findings provide actionable guidance for benchmark design, RAG system development, and LLM training. Code is available at https://github.com/OuchengLiu/LLM-Misalign-Climate-Change.