Perception is for observation, Investigation is for deduction.
Some of this answer will be observations on how Wizards has done it so far and some of this will be logic, and some of it will be mechanics.
First for the headline question. It depends. when to use Investigation, and when to use Perception is not entirely clear yet, and I'm hoping we'll get more guidance in the DMG late this year. The guidance from the rules is that the two skills mostly seem to differ in the methods by which they are found.
Depending on the exercise, either, or both of the skills may be used.
For passive checks, you're almost always looking at Perception being the skill of record. While all skills can be used passively, some skills make more sense than others. Perception is the hallmark passive skill, whereas investigation makes less sense as a passive skill.
If the character is alert to the possibility of hidden objects/traps, but not actively searching, he's using passive perception.
The guidance for this seems to be (though we can't confirm yet), that the DC for actively looking for something is regularly about 5 less than the DC for passively looking for something (or sometimes actively looking always reveals it).
So here's how I would play it. Traps can be noticed with perception passively (usually DC 15). If the PC is looking, ask them how they are looking. If it's observational, then use Perception. If it's deductive, use Investigation. When they are searching for items, again, either skill is appropriate. This is somewhat counter to how WOTC has written adventures so far. They are always written to use perception to notice traps. Passive with a higher DC and active with a lower.
Investigation also has broader uses such as when you are trying to track clues, or put something together. It's also a great "roll for a hint" kind of skill if your PCs get stuck and need some help figuring out what to do next.
Passive perception is exactly that, passive. It's what the PCs are always using when not actively searching for something and doesn't use a roll of the die.
To determine if you should use passive perception or allow a player to roll, listen to what they say their PCs are doing. If they say they are standing watch, keeping an eye out or something similar, they are actively searching so they can make a wisdom (perception) check, otherwise they are using their passive perception.
Although, it's kind of an experience call as the DM and depends on what the PCs are actually doing and whether or not the DM gives them the benefit of the doubt. In the case of the video, one guy said he was keeping a look out, thus actively searching and two of them were scouting off to the side of the road so the DM ruled they were also searching, rather than just wandering off into the bushes for no real reason, and allowed those three PCs an active wisdom (perception) check, whilst the others would be using their passive perception.
Best Answer
Yes. Passive perception "supersedes" active perception by acting as a floor.
The party scout was correct. Passive perception does not "turn off" when you are actively searching. It only stops if you are unconscious. You always notice anything that hasn't beat your passive perception score — even if you aren't actively searching.
You can also take actions to actively search, giving you an opportunity to sense things you haven't already noticed.
For example, if your passive Wisdom (Perception) score is 15, and a monster is lurking with a Dexterity (Stealth) roll of 14, you notice them without rolling. If the monster got a 15 or higher, though, they've beaten your passive senses. Let's say that the monster has a modifier of +3 and rolled 16 on the die, for a result of 19. In this case, on your turn, you can take actions to actively search, for which the DM can call for Wisdom (Perception) rolls. If you exceed the monster's total, you've noticed it. If you get the same total or less (19, here) the situation remains the same and the monster is undetected. For that first monster — the one with a 14 total Dexterity (Stealth) check — the situation also remains the same, which is that you've already noticed them. They aren't somehow "de-noticed".
Jeremy Crawford explains in the Sage Advice section of this podcast, starting at about 15:09.
(Bold added to highlight the key points; italics intended to represent emphasis in the speech.)