Intelligence/Investigation vs Wisdom/Perception: Wisdom reflects how attuned you are to the world around you and represents perceptiveness and intuition. Intelligence measures mental acuity, accuracy of recall, and the ability to reason.
I use Wisdom and Perception rolls for discovering if, and Intelligence rolls (of all types) for discovering what.
Discovering if tells me whether something is realized or perceived. It doesn't generally explain the meaning of what is discovered. A perception check will allow a player to notice the footprints in the rug, or hear the breathing. It will give facts, not conclusions.
Discovering what gives more substantive answers. An intelligence/investigation check would allow a character to deduce that those footprints are an invisible creature, or are an illusion, or are from a ghost, or are whatever else they could be.
Investigation is used to determine things that have layers. A knowledge skill (arcana, history, religion, nature) will give you facts, but investigation is the primary skill for figuring something out, connecting the dots, and putting together information.
Passive Rolls: Passive checks can represent the average result for a task done repeatedly, such as searching for secret doors over and over again, or can be used when the DM wants to secretly determine whether the characters succeed at something without rolling dice, such as noticing a hidden monster. There are specific rules for hiding that are unique to hiding and passive perception - stealth rolls are opposed by passive perception, explicitly (unless the target is actively using their action to look).
I use passive scores as a floor for ability scores when a PC is not focused on something else. If they are focused on something, then there is no floor on a roll. If you're trying to jump over a pit and focused only on doing so, you have a higher floor on it than if you did it in the middle of combat. If you're negotiating a deal over a cup of tea, you're going to have a higher floor on your persuasion than if you plead with someone to stop attacking you.
Clarifying note: A lot of situations can be approached with an array of options. I listen to what the characters are doing when they're telling me something (as opposed to when we're using passive perceptions, etc...) to determine the roll to be made. It is the character actions that drive whether wisdom or intelligence roll.