One can rigorously prove that pyramid schemes cannot run forever, and that no betting system with finite monetary reserves can guarantee a profit from a martingale or submartingale.
But there are countless examples of people who have suffered monetary loss due to their lack of awareness of the rigorous nature of these non-existence proofs. Here is a case in which having a non-rigorous 99% plausibility argument is not enough, because one can always rationalise that "this time is different", or that one has some special secret strategy that nobody else thought of before.
In a similar spirit: a rigorous demonstration of a barrier (e.g. one of the three known barriers to proving P != NP) can prevent a lot of time being wasted on pursuing a fruitless approach to a difficult problem. (In contrast, a non-rigorous plausibility argument that an approach is "probably futile" is significantly less effective at preventing an intrepid mathematician or amateur from trying his or her luck, especially if they have excessive confidence in their own abilities.)
[Admittedly, P!=NP is not a great example to use here as motivation, because this is itself a problem whose goal is to obtain a rigorous proof...]
