Computer models depend on the fact that mathematical equations can describe the physical changes that govern the weather, just as equations describe movements of the solar system well enough for solar and lunar eclipses to be predicted years, even centuries, in advance.
Equations describing the solar system are complicated.
Still, mathematicians and astronomers have been able to predict eclipses for centuries. The atmosphere's equations are much more complex. Solving them had to wait not only for more knowledge, but also for computers. The atmosphere is just too complex, among other reasons. When researchers began developing the first computer models for the earth's atmosphere in the 1950s, they worked with computers that were extremely limited compared with today's.