Wisdom is knowing what to do in most situations. Intelligence is knowing what to do in a few, spectacularly.
That's the distinction worth making. "Wise" people achieve high average outcomes across contexts. "Smart" people perform brilliantly in narrow domains, often by holding contradictory thoughts simultaneously.
Two popular explanations miss the mark:
Wisdom applies to human problems, intelligence to abstract ones. This fails immediately. The engineer who chooses failure-resistant structures demonstrates wisdom without touching human problems.
Wisdom comes from experience, intelligence is innate. But people aren't wise proportional to their experience. Years don't automatically create sound judgment.
Wisdom is harder to pin down than intelligence because it encompasses a grab-bag of qualities: self-discipline, experience, empathy. It's not one skill but consistent good judgment across situations.
The wise person knows what to do on average. The smart person knows what to do in specific cases, sometimes brilliantly. Both are valuable. They're not the same thing.