Answer:
Computers are used for interpolating data, and analyzing very complicated data.
Explanation:
In forecasting we need to look at a lot of weather variables (temperature, dew point, pressure and wind speed to name a few). We only collect this data from weather observation sites which can be tens to hundreds of kilometers apart. In order to determine these values at locations between the observation sites we interpolate.
When you are forecasting you need look at a lot of data and determine what is going on to determine what is going to happen next. If humans did all the interpolating themselves for all the data they would never be able to issue the forecast in the time that is expected. Furthermore, now that we have very complicated values to analyze (vorticity for example) there is no way that a human can calculate the effect of all the various factors in play. A computer has to do it.
Computers are also required for using weather radar, and yes to a lesser extent satellite data.