This is a major point of confusion for people who are new to the concepts, and I'm glad someone has taken the time to explain it in depth. And it's not just TSP that it happens with. Almost every popular example of an NP-complete problem I've seen is actually an NP-hard optimization problem corresponding to an NP-complete decision problem.
The bad part is that if you don't know better, you'll think "Wait, how the hell would you verify a correct answer to this without searching for an even shorter path?" And you'll be right, but you'll feel like you're missing something. Giving examples of CS that make people feel like they fundamentally don't understand CS is probably a Bad Thing.
The bad part is that if you don't know better, you'll think "Wait, how the hell would you verify a correct answer to this without searching for an even shorter path?" And you'll be right, but you'll feel like you're missing something. Giving examples of CS that make people feel like they fundamentally don't understand CS is probably a Bad Thing.