Actually, there are only a handful of variables in a weather simulation. For a typical cloud-scale simulation you have the three components of wind, moisture, temperature, pressure, and precipitation variables. Say, 13 variables. That is not why you need supercomputers.
The reason you need supercomputers to do weather simulations is all about resolution, both spatial and temporal. Weather simulations break the atmosphere into cubes, and the more cubes you have, the better you resolve the flow. All weather simulations are underresolved; to properly model the turbulent flow in the atmosphere you need to get down to cubes that are roughly a centimeter on a side. As you double the resolution (halve the length of each of the four lines that makes up a cube face) you require eight times as many cubes. In weatherspeak, we talk about gridpoints instead of cubes where it's understood that each gridpoint represents the center of one of these cubes. In the computer model, they are represented as three dimensional floating (or double precision) point arrays. So take a 3D array and double the number of calculations on each of the thee for: loops, and you've got eight times as many calculations and eight times more memory required.
And it gets worse. When you double the resolutions, you need to halve the time step. Weather models step forward in time in discrete intervals, and now in addition to more calculations for each time step (eight times as many for doubling the resolution in three dimensions) now you need to go in steps that are half as large. This means 16 times more calculations, and eight times as much memory, to double the resolution.
And many of the calculations that are being made in the innermost loop involve things like divides, non-integers powers, square roots, etc... expensive calculations. And then because it's a massively parallel simulation, you have to do internode communications - which adds overhead and can be rather a bother. Then there's the hundreds of TB of data the model is dumping to disk. Now let's render that, shall we? Somebody call Pixar.
I am working on a project to simulation a thunderstorm which will produce a tornado in a "natural" way. The tornado needs to be adequately resolved. This simulations will have grid spacing of 10 meters. It requires a computer which hasn't been fully built yet (Blue Waters, in Urbana, google it). The time step will be 0.01 seconds, and the model will run for two hours of model time. It will take days of wallclock time. Keep in mind this model will have a physical domain not much bigger than about half the area of Oklahoma. Imagine global climate modeling now, and now you're talking 4 km resolution being all you can do.
This is why we need supercomputers to do high resolution weather simulations.