Large Language Models (LLMs) often generate overly cautious and vague responses on sensitive topics, sacrificing helpfulness for safety. Existing evaluation frameworks lack systematic methods to identify and address specific weaknesses in responses to sensitive topics, making it difficult to improve both safety and helpfulness simultaneously. To address this, we introduce FINEST, a FINE-grained response evaluation taxonomy for Sensitive Topics, which breaks down helpfulness and harmlessness into errors across three main categories: Content, Logic, and Appropriateness. Experiments on a Korean-sensitive question dataset demonstrate that our score- and error-based improvement pipeline, guided by FINEST, significantly improves the model responses across all three categories, outperforming refinement without guidance. Notably, score-based improvement -- providing category-specific scores and justifications -- yields the most significant gains, reducing the error sentence ratio for Appropriateness by up to 33.09%. This work lays the foundation for a more explainable and comprehensive evaluation and improvement of LLM responses to sensitive questions.