Question

Surely calculus is the ultimate treasure trove for such examples. In antiquity, the Egyptians, Greeks, Indians, Chinese, and many others could calculate integrals with a pretty good degree of certainty via the method of exhaustion and its variants. But it is not without reason that Newton and Leibniz are credited with the invention of calculus. Because once you had a formalism- a proof- of the product rule, chain rule, taylor expansions, calculation of an integral- in fact, once you had the formalism in hand to make such a proof possible- then with that came an understanding, and from that sprung the most powerful analytic machine known to man, that is calculus. Without a formalism, Zeno's paradox was just that- a paradox. With the concept of limits and of epsilon-delta proofs, it becomes a triviality.
Thus, in my opinion, proof is important in that it leads to mathematics. Mathematics is important in that it leads to understanding patterns, and patterns govern all of science and the universe. If you can prove something, you understand it, or at least "your concepts understand it". If you can't prove it, you're nothing more than a goat, knowing the sun will rise in the morning from experience or from experiment, but having not the slightest inkling of why.
The specific example, then, is "calculating integrals" and "solving differential equations".

With the reader's indulgence, an example of a mathematical proof saving lives. My friend's mum is an aeronautical engineer at a place which designs fighter jets. There was some wing design, whose wind resistance satisfied some PDE. They numerically simulated it by computer, and everything was perfect. My friend's mum, who had studied PDE's seriously in university and thought this one could be solved rigourously, set about finding an exact solution, and lo-and-behold, there was some unexpected singularity, and if wind were to blow at some speed from some direction then the wing would shear off. She pointed this out, was awarded a medal, and the wing design was changed. Lives saved by a proof. I'm sure there are a thousand examples like that.