Which skill should be used for finding secret doors or traps: Perception or Investigation?

If I only look at the SRD, it seams pretty cut and dry; all the examples for Wisdom (Perception) are creatures and all the examples for Intelligence (Investigation) are objects, but if I look at the Player's Handbook, page 178, the green box example is using Perception for secret doors or traps.

Specifically, under Investigation:

deduce the location of a hidden object,

And the green box:

When your character searches for a hidden object such as a secret door or a trap, the DM typically asks you to make a Wisdom (Perception) check.

Which is correct?

Seeing and Finding are different

The rules on these two skills are blurry and have a significant amount of overlap. That means the exact correct answer is "up to the DM". You make the check the DM decides is more appropriate. I will detail the way I choose between them, this method is also used by every DM I have played with. It also seems to be the method used by Matt Mercer, though I haven't asked him to be sure.

Perception is for seeing

Your Wisdom (Perception) check lets you spot, hear, or otherwise detect the presence of something. It measures your general awareness of your surroundings and the keenness of your senses. - PHB

Perception checks are based on your senses. Can you see, hear or feel something that might give away the location. An example of how this can be used to find hidden doors:

You are standing in the centre of the room, you glance around to notice traps, triggers and anything else of interest. You feel a faint breeze blowing from the bookshelf in the corner.

This is a Perception check. It is a measure of your ability to notice things by sight or sense.

Investigation is for searching

When you look around for clues and make deductions based on those clues, you make an Intelligence (Investigation) check. - PHB

Investigation checks are about using logic and reasoning to figure out where the trap or secret door may be. This is a narrative difference rather than a mechanical one. An example description a DM may give:

You hunt around the room, methodically checking every shelf, cupboard and wall sconce for a hidden switch. You find scratch marks on a certain section of floor and deduce that the bookshelf must swing out.

This is an Investigation check. It is a measure of your ability to think logically and methodically locate something. You are using the clues to figure out where the secret door lies.

Finding a Hidden Object Green Text

I believe most of the confusion is coming for the block of green text under the Perception section of ability checks. In that section it has the text below in relation to Perception checks (emphasis mine):

... looking at the walls and furniture for clues, [...] you were opening drawers or searching the bureau... - PHB

This text seems to contradict the non-green-text descriptions for these two skills. I can't speak to why WotC choose to include this but I can try to explain the difference in spite of it.

Investigation is your ability to think and reason based on evidence. Perception is your ability to find that evidence with your senses. A DM could possibly call for both checks for a particularly difficult situation. More likely is you decide which is more difficult, finding the evidence or understanding it. Then call for the appropriate check.

This method is how I have always handled it at the table, as does my DM. The players and DM always seems to have a good idea of which situations require which check and it has never caused an issue.

Perception is used for senses

"You walk down the hallway and you see a row of tiny holes along the bottom of the right-hand-side wall, what do you do?"

Primary sensations you can see or hear or smell or touch or feel in the air or similar. PHB p 178.

Investigation is for knowledge & reason

"Yes, the lack of dust in a quarter circle in front of the wall does look like a secret door has been recently opened here, and then closed again."

Knowledge of architecture, design, common sense… and reason to draw conclusions from the clues you've used perception to gather. PHB p 178.

Active vs Passive

When you're just moving in the dungeon with your eyes, ears or nose open, sometimes you (as a character) perceive things even without you (as that character's player) asking to actually look. As an example, sometimes the character sees a tripwire even without the player going "I'm going to look to see if there's a tripwire" in every room.

Using a skill passively means walking around with a permanently rolled 10 (or 15 if you have advantage), modified by your bonus, and then compared to the various DCs. For example, a particular tripwire might have a passive perception DC of 15 and then anyone with passive perception 15 or higher will see it without actively looking. A set of footprints might look weirdly overrepresented by steps leaving a small, closed room, and anyone with a passive investigation of 15 or higher will realize that without actively pondering it.

Suggestion

Passive perception is OK. It allows you to selectively describe the dungeon, normally to everyone and then to high-perception characters add things like "[character name], you see a tiny crack along the left-hand-side wall". Some modules have DCs already listed. When writing homebrew modules, or prepping modules that don't have DCs listed, write the DC down beforehand and stick to it. One idea is to roll 2d6+8 (or 4d6+1, or 4d3+7) to get a DC that's centered around 15 [the standard, "medium" difficulty] but doesn't make any single perception score actively best (i.e., no "the optimal passive perception to shoot for is 15, it's worthless to go lower and a waste to go higher"). Again, do that roll beforehand so you can write it down & stick to it.

Active perception, and both passive and active investigation, interacts weirdly with player skill. Therefore, don't use them. (It's true to some extent to passive perception as well.)

"I open the drawer and lift up the clothes, do I see the key there?"; this is Perception. PHB p 178 has a similar example.

But if the prep says that the figurine is actually in there, should you then really have to roll to see if you can see it? At some point it becomes ridiculous. "I grab my sword!" "OK, roll to see if you can feel your own sword."

The PHB's example points out that if the key is in the drawer under the clothes, looking at the walls of the room would give you no chance of success. So obviously player skill does enter into the equation.

Using these character skills and player skills together leads to one of two things.

So you have four options:

• The character skill becomes a lifeline for when player skill is insufficient. You let them find the key without them looking under those clothes, and if they don't look in the right place you let a high perception roll give them another chance to find it.


• The character skill becomes an additional hurdle to jump even when player skill is high. You don't let them find the key even if they look in the exact right place until they've also made a perception check.


• The player skill is ignored. Don't dwell on precise description until you've seen the outcome of the die rolls. It becomes "Roll to solve puzzle."


• The character skill is ignored. Don't roll these skills and instead rely solely on player skill.

Decide (together with the group) which of these four interactions you prefer and apply it consistently. (As a side suggestion, if you do go without using a lot of skill rolls---for example, if you are also not using Persuasion, Deception, Intimidation and Insight, for the same reason---is the Ability Check Proficiency variant in the DMG on p 263.)

As an example, our group uses the fourth option and that ability check variant. Last week, we played two four-hour sessions and in those eight hour sessions there was a total of four ability checks. (One strength check to grapple, two strength tests to budge open doors, and one int check to recognize that a book about golems was specifically only about clay golems.) It's perfectly fine to play D&D without every action or statement being accompanied by an ability check.

Pouring out flour to see trip wires, pouring water to see trap doors.

Just describe the room and ask "what do you do?"

What's fun to roleplay out vs what's fun to roll is different between groups. You might want to roll for trap finding if you hate pixel hunting but then play out social interactions, or you might want to play out trap searching if you love exploration but then just roll to quickly resolve social interactions if that's what you want to do.

Exceptions

When is that fourth option no good?

• When you are winging it and you don't know if the key is in the drawer or not because you don't have that in the prep


• When the group has no desire for player skill but instead are highly invested in finding out "what would the character have done in the situation?"


• When there's a large time skip: "OK, we search through every room on the top first two floors. Three days later, have we found anything?"


• When you desire a high differentiation between characters; a low-int character being played ignorantly and a low-wis character being played obliviously.


• When you desire a hilarious amount of slapstick and mistakes. "Oh, wow, I can't believe my character didn't see that tripwire even after I looked so carefully, hahaha, well, time to roll up a druid, I guess."

But if none of those exceptions apply to you, you might want to want to forgo these types of checks. If you still want to use these skills, hopefully the differentiation at the top of the answer can be useful to you.