In the coded aperture snapshot spectral imaging (CASSI) system, the coded and compressed single-channel measurements need to be reconstructed into hyperspectral cubes. Existing discriminative models reconstruct the spectral cube by optimizing the mean squared error (MSE) between the ground truth and the predicted image, employing peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM) as metrics to gauge the quality of reconstruction. However, these indicators often possess significant limitations in mimicking human visual perception and in discerning the impact of image distortions on perceived visual quality. In this article, a new model named CASSIDiff is proposed to reconstruct CASSI measurements, achieving advanced results in perceptual loss-based evaluation metrics such as learned perceptual image patch similarity (LPIPS) and Fréchet inception distance (FID). The diffusion model, which enjoys high accuracy and reliability in generative tasks, is used for the first time for the hyperspectral reconstruction task. A feature fusion mechanism based on discrete wavelet transform (DWT) is used to weaken the noise interference effect in the conditional diffusion model. Considering the interspectra similarity and long-range dependencies of hyperspectral data, the spatial-spectral attention mechanism is also introduced. Experiments show that CASSIDiff not only outperforms most existing algorithms in simulation datasets but also shows robustness to real data published and collected in our home-built CASSI system. The code and models are publicly available at: https://github.com/YifanSi/CASSIDiff.