User interests can be viewed over different time scales, mainly including stable long-term preferences and changing short-term intentions, and their combination facilitates the comprehensive sequential recommendation. However, existing work that focuses on different time scales of user modeling has ignored the negative effects of different time-scale noise, which hinders capturing actual user interests and cannot be resolved by conventional sequential denoising methods. In this paper, we propose a Long- and Short-term Interest Denoising Network (LSIDN), which employs different encoders and tailored denoising strategies to extract long- and short-term interests, respectively, achieving both comprehensive and robust user modeling. Specifically, we employ a session-level interest extraction and evolution strategy to avoid introducing inter-session behavioral noise into long-term interest modeling; we also adopt contrastive learning equipped with a homogeneous exchanging augmentation to alleviate the impact of unintentional behavioral noise on short-term interest modeling. Results of experiments on two public datasets show that LSIDN consistently outperforms state-of-the-art models and achieves significant robustness.