In the Sunless Citadel module, from Tales from the Yawning Portal, Passive Perception is mentioned about half a dozen times, but Passive Investigation isn't mentioned once.
However, both passives are increased by +5 by the Observant feat, and both are included when it comes to the Inquisitive rogue subclass in XGtE. In most mechanical and class mentions of Passive Perception vs. Passive Investigation, it seems they are supposed to be equals.
Passive Perception is used as the defensive stat for enemies attempting to use Stealth, but no such claim is made for Passive Investigation.
So when is Passive Investigation used, beyond being an automatic permanent minimum roll for an unmade active investigation check?
Personally, I like to use them as reactionary/defensive stats that the DM rolls against, for things that a character can miss (somebody passing by, a drop of water falling from the ceiling, knowing the intended target of a shot fired from an assassin). This is not, however, an official use of the passive skills. I'm interested in regular uses of the passive skills for the sake of this question.
Best Answer
The distinction between perception and investigation isn't that great
As I've seen it played out at table, passive Intelligence (Investigation) checks are most often called out by the DM when hidden things (secret doors, items hiding in plain sight, illusions, etc) are in the area that the PC can see or hear.
That said, most DM's I have played with default to Perception for a great deal of 'you notice this' more than any Intelligence-based skill. (A notable exception being our ToA DM, who uses a variety of ability based checks depending on the situation).
The most common applications I have seen in game is a secret doors detection (a player had taken the Observant feat), the 'feet sticking out from under the curtains' deal (yeah, our PC spotted the thief who had run away) and a whole gamut of things that "just don't look right." Two instances stand out
The old 'you notice a lack of a shadow' during an encounter with what was (we discovered soon after) an illusory creature.
What I have also seen that fits very well is the use of passive Intelligence based checks that let the DM just advise one of the players "you recall (X)" without interrupting play for a roll.
For a good illustration of the distinction between the two different ability checks, see @DaleM's answer.
About passive checks in general
As my friend Tanarii has suggested here, passive checks ...