They'll find out it's really not, when they inevitably meet a foolish highly-intelligent person doing unwise things...

Intelligence: Here is how you can remove this fence.

Wisdom: Should we really be removing this fence?

My fighter is somewhat wise but not so smart. At least he knows he isn't the smartest, thus he writes everything down in a journal he deems important. He doesn't come up with the most interesting or cunning solutions. But he knows that good preparation, a solid formation and sharp weapons can solve many problems and avoid pitfalls.

The opposite (smart but not wise) would be someone who is book smart or even a smart-ass. Probably coming up with all sorts of crazy and interesting ideas, but doesn't necessarily judge situation too well and might quickly discard simple but effective solutions. They might be quick minded and suddenly do high risk/reward kind of moves that either save the day or fail spectacularly.