Recent salient object detection (SOD) models predominantly rely on heavyweight backbones, incurring substantial computational cost and hindering their practical application in various real-world settings, particularly on edge devices. This paper presents GAPNet, a lightweight network built on the granularity-aware paradigm for both image and video SOD. We assign saliency maps of different granularities to supervise the multi-scale decoder side-outputs: coarse object locations for high-level outputs and fine-grained object boundaries for low-level outputs. Specifically, our decoder is built with granularity-aware connections which fuse high-level features of low granularity and low-level features of high granularity, respectively. To support these connections, we design granular pyramid convolution (GPC) and cross-scale attention (CSA) modules for efficient fusion of low-scale and high-scale features, respectively. On top of the encoder, a self-attention module is built to learn global information, enabling accurate object localization with negligible computational cost. Unlike traditional U-Net-based approaches, our proposed method optimizes feature utilization and semantic interpretation while applying appropriate supervision at each processing stage. Extensive experiments show that the proposed method achieves a new state-of-the-art performance among lightweight image and video SOD models. Code is available at https://github.com/yuhuan-wu/GAPNet.