General circulation models (GCMs) struggle to accurately simulate precipitation, particularly extremes and the diurnal cycle, which are crucial for both human activities and natural processes. Although hybrid models combining machine learning and physics offer a promising avenue to improve the simulation of precipitation, they have yet to outperform existing GCMs. Here, we present a hybrid model built on the differentiable NeuralGCM framework. This differentiability facilitates direct training on satellite-based precipitation observations, unlike previous attempts at hybrid models that relied on high-resolution simulations as training data. Our model runs at 2.8° resolution and, in the context of climate, demonstrates substantial improvements over existing GCMs, the ERA5 reanalysis, and a global cloud-resolving model in simulating precipitation. In the context of mid-range precipitation forecasting, it outperforms the ECMWF ensemble. This advance paves the way for more reliable simulations of current climate and demonstrates how training on observations can be used to improve GCMs.