The biggest, most common mistake I see in poker goes like this:
Say you flop a pair and the flop has 2 of a suit and a straight draw. You are betting your pair and Dude is calling. You assume he is on the draw. After the turn and river, no flush or straight comes. He checks. You figure his hand is a bust. You bet.
Well, If you are correct and he has been busted, he won't call, so you get the current pot (upside). Downside is that he could raise you with something like a back door set and you lose 2 bets. If you simply check and show, the best case scenario is you take the pot (same upside scenario as when you bet). The downside is that you lose, and lose no more money (2 bets worth of a better scenario than if you bet).
If the upsides are equal, but one downside is worse than the other, then in Game Theory parlance, you have a "Dominating Decision" and the decision is clear.
The only situation that I can think of where you would be better off betting is if Dude had a middle pair with his draw and figured that it was worth the call on the end to a possible bluff from you. Not only is this scenario rare, Dude's decision to call in this scenario is rare. Thus, it is not worth really factoring into the decision.
You have nothing to gain by betting into an apparent busted drawing hand, so don't do it.