Many examples that have been given are of statements that one could at least formulate, and conjecture, without rigorous proof. However, one of the most important benefits of rigorous proof is that it allows us to step surefootedly along long, intricate chains of reasoning into regions that we previously never suspected existed. For example, it is hard to imagine discovering the Monster group without the ability to reason rigorously.
In any other field besides mathematics, as soon as a line of abstract argument exceeds a certain (low) threshold of complexity, it becomes doubtful, and unless there is some way to corroborate the conclusion empirically, the argument lapses into controversy. If you are trying to search a very large space of possibilities, then it is indispensable to be able to close off certain avenues definitively so that the search can be focused effectively on the remaining possibilities. Only in mathematics are definitive impossibility proofs so routine that we can rely on them as a fundamental tool for discovering new phenomena.
The classification of finite simple groups is a particularly spectacular example, but I would argue that almost any unexpected mathematical object—the BBP formula for $\pi$, the Lie group $E_8$, the eversion of the sphere, etc.—is the product of a sustained search involving the systematic and rigorous elimination of dead end after dead end. Of course, once an object is discovered, you might try to argue that mathematical rigor was not really necessary and that someone could have stumbled across it with a combination of luck, persistence, and insight. However, I find such an argument disingenuous. Mathematical rigor allows us to distribute the workload across the entire community; each reasoner can contribute his or her piece without worrying that it will be torn to shreds by controversy. Searches can therefore be conducted on a massively greater scale than would be possible otherwise, and the productivity is correspondingly magnified.