ExploderWizard
Hero
Intelligence (Investigation) is for Active search for things you don't know are there or can't perceive without interaction.
Wisdom (Perception) is for Passive Search and for Active Search of things you can already perceive.
Basically
Perception is for searching in plain sight.
Investigation is for search in hidden view.
Like the PHB say, you can't find a key in a drawer with Perception unless you say "I open the drawer" to open the drawer.
But you can find the key with Investigation because rolling high enough "makes you open the drawer".
Also Investigation is the "logical step search".
Perception tells you that the noble's shoes are dirty.
Investigation tells you (if pointed out via Perception or interaction) that the dirt is fresh, what type of dirt it is, and that noble is a dirty liar and he did go outside during the ball..... ARREST THAT MAN AND HOLD HIM FOR FURTHER QUESTIONING!!!.
Players don't ask to make checks or "use skills" at my table. They describe what they want to do and then I tell them the result, sometimes asking for a check if I think their goal and approach relative to the situation has an uncertain outcome. In a game where the DM decides on success, failure, or uncertainty, players shouldn't really want to roll if you think about it. They should just want outright success and roll only when necessary.
The only stumbling block to this very wonderful and awesome way of doing things are players that learned how to play in WOTC editions of the game. The general rules-first approach of these systems impresses upon the player that nothing of consequence or meaning takes place without a die roll and that anything is doable with a high enough result. The formula being -1) find out what skills are applicable, 2) find the characters with the highest modifiers to those skills, 3) have these characters perform the task, 4) rinse, repeat.
Actually interacting with the environment instead of the rules is even trickier when the game is run by a DM who was taught only WOTC D&D. If a die roll as a gatekeeper of anything meaningful happening is ingrained in the DM then players calling out skill checks is the only way anything is happening and thats kind of sad.