A very well-known example is atomic weapons. It is much easier to build a device and test it than it is to simulate it: it took a very long time before simulations of existing designs and small variations on them became convincing enough that people had any faith in them at all, and I believe that we can really only use simulations now because it became clear that small variations on existing designs were adequate, and radical innovations were no longer needed (no-one wanted suitcase nukes or hundred-megaton devices to exist). Even now there are many physical tests of critical aspects of weapon systems (some of this sort of thing goes on at the National Ignition Facility).
Of course full tests of nuclear weapons have several undesirable side-effects, so people were eventually pushed into simulations combined with physical tests of things that were too hard to simulate.
More generally almost any complex system -- which is almost any system -- is generally either ludicrously hard or completely impractical to simulate numerically.
More generally still, if you are just doing simulations, you're not actually doing physics: you're testing a model you've made, which can be interesting in itself, but to do physics you need to be comparing the model with the world.