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Out-of-distribution (OOD) detection is essential for model trustworthiness which aims to sensitively identify semantic OOD samples and robustly generalize for covariate-shifted OOD samples. However, we discover that the superior OOD detection performance of state-of-the-art methods is achieved by secretly sacrificing the OOD generalization ability. Specifically, the classification accuracy of these models could deteriorate dramatically when they encounter even minor noise. This phenomenon contradicts the goal of model trustworthiness and severely restricts their applicability in real-world scenarios. What is the hidden reason behind such a limitation? In this work, we theoretically demystify the ``\textit{sensitive-robust}'' dilemma that lies in many existing OOD detection methods. Consequently, a theory-inspired algorithm is induced to overcome such a dilemma. By decoupling the uncertainty learning objective from a Bayesian perspective, the conflict between OOD detection and OOD generalization is naturally harmonized and a dual-optimal performance could be expected. Empirical studies show that our method achieves superior performance on standard benchmarks. To our best knowledge, this work is the first principled OOD detection method that achieves state-of-the-art OOD detection performance without compromising OOD generalization ability. Our code is available at \href{https://github.com/QingyangZhang/DUL}{https://github.com/QingyangZhang/DUL}.