If differential equation theory was known and also studied by Galileo, so why he didn't manage to discover a normal distribution (its discovery had to wait for Laplace and Gauss)?
Even aside from the fact that Galileo knew nothing of differential equations, or derivatives for that matter (he lived before Newton and Leibniz), and that the normal distribution was not discovered by Laplace and Gauss but by De Moivre, why the connection? De Moivre discovered the bell curve not by solving differential equations but looking for a good approximation to binomial distributions with large n, Bernoulli's formula with binomial coefficients was not very practical for calculations. And Galileo could not do that either because even the simplest cases of the binomial distribution (or of any statistical distributions, for that matter) were not considered until the Fermat-Pascal correspondence 12 years after his death.
Laplace did try to derive the error curve using differential equations, but the curves he derived that way had cusps or vertical asymptotes at the origin and did not even look like the normal distribution, see Stahl's Evolution of the Normal Distribution.