Precipitation changes can affect society more directly than variations in most other meteorological observables, but precipitation is difficult to characterize because of fluctuations on nearly all temporal and spatial scales. In addition, the intensity of extreme precipitation rises markedly at higher temperature, faster than the rate of increase in the atmosphere's water-holding capacity, termed the Clausius-Clapeyron rate. Invigoration of convective precipitation (such as thunderstorms) has been favoured over a rise in stratiform precipitation (such as large-scale frontal precipitation) as a cause for this increase, but the relative contributions of these two types of precipitation have been difficult to disentangle. Here we combine large data sets from radar measurements and rain gauges over Germany with corresponding synoptic observations and temperature records, and separate convective and stratiform precipitation events by cloud observations. We find that for stratiform precipitation, extremes increase with temperature at approximately the Clausius-Clapeyron rate, without characteristic scales. In contrast, convective precipitation exhibits characteristic spatial and temporal scales, and its intensity in response to warming exceeds the Clausius-Clapeyron rate. We conclude that convective precipitation responds much more sensitively to temperature increases than stratiform precipitation, and increasingly dominates events of extreme precipitation.