If you fail a check against a henchman who is a monster, you take damage just as you would facing any monster, in addition to any other penalties that the card may give you if it's undefeated.
What you probably saw in that video is that if you defeat the henchman in a particular location, it's banished back to the box, and most of the time you are given an opportunity to close the location. If you fail the When Closing requirement at the location, you don't close it, but now there generally isn't another card in the location that grants you a second chance to close. The only way you'll close that location is to grind through all the cards that are left at the location. Then before you end your turn, you get a chance to close the empty location.
If you're fighting a villain, you're at a location, but that location isn't closed. (It would be closed if you had defeated the villain, but...) This means that there is a location for the villain to escape to: the location where you found it. Shuffle the villain card back into that location. No blessings need to be taken off the "clock". You got off easy, and now you have the villain cornered in one location. You might decide to back away slowly, leave that location for a while and permanently close some of the other locations so you can bring more characters into the fight.
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Henchmen are going to be either monsters or barriers. If they're monsters, then you take damage from them the same way you would any other monster. If they're barriers, they'll have some other description of what happens if they're undefeated. If you lose against a henchman, you shuffle it back into the deck it came from, like any other monster or barrier.
What you're concerned about comes into play when you defeat a henchman but fail to close the location. Most (but not all!) henchmen say that if you defeat them, you may immediately attempt to close the location they came from. (Note that this does not work with summoned henchmen, because they did not come from a location.) If you defeat the henchman, it is banished. If you then fail to close the location, it cannot be permanently closed until it is empty. So often a good point of strategy is to chase the villain to that location, or to plan to temporarily close it when you encounter the villain.
Okay, so when a villain is not defeated. Villains, again, are either monsters or barriers and behave accordingly in terms of damage. The location the villain is in does not permanently close, and the villain escapes to any open locations (not those that were temp closed or that are permanently closed). But the blessings that are shuffled with the villain do not come from the box---they come from the blessings deck. So you lose turns. It's always better to defeat villains!
Hope that helps!
EDIT: Alas, ninja'd!
Henchmen have a Type box above their Check to Defeat: It should say either Type: Monster or Type: Barrier. The henchman is a monster if it says it's a monster, and a barrier if it says it's a barrier. If you fail a check to defeat a monster, you take combat damage equal to how much you failed by. Barriers don't have any implicit damage for failing, but the henchman may have other powers that apply if undefeated.
For the villain scenario, by the Rules As Written you win the scenario despite failing to defeat the villain. This can only happen if you have every other location closed and you're fighting the villain in a location like the Abyssal Rift while it is on the "always temporarily closed" side (remember you can never attempt to temp close the villain's location so it needs to already be that way). I don't believe there was ever an official resolution for this, but what I would do is just shuffle it back into the Abyssal Rift in that case.
Note skizzerz is answering the exceptional situation in which you are fighting a villain at a location which is already closed. This almost never happens. (Although it is what the OP described.)
Typically the situation is as Axoq says -- the villain's location is open, and he or she can escape there.
Chief Technical Officer
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I suspect the OP was asking what happens when all *other* locations are closed, but you don't defeat the villain.
The answer is that you are already going through the steps of "Encountering a Villain," and you need to keep going through them. If you have failed to defeat the villain, that means you just finished the step headed "Encounter the Villain." Because you didn't defeat the villain, nothing in "If You Defeat the Villain, Close the Villain’s Location" applies. So you move on to "Check to See Whether the Villain Escapes." The first sentence says "If any locations are not closed, the villain escapes." Since the current location is the only one that isn't closed, that's where the villain will escape to.