If there wasn't passive perception, each character would be rolling perception automatically whenever a hidden creature was around, as in 3E*. That is how the passive perception system makes stealth more viable. Spotters are forced to "take 10" unless devoting an action to find something.
This is not entirely correct. According to passive perception in PHB, it's purpose is to simulate making lots of rolls over a long period of time, averaging in a roll of 10. So it potentially "saves time" rolling repeatedly (if as a DM you permit re-rolling of ability/skill checks), and to prevent the occasional inadvertant player tipoff to an impending trap, ambush, whatever (by virtue of you rolling a perception check for PCs behind your screen - although frankly this should not really tip anyone off if you make rolls behind your screen not infrequently, or if the ambush etc is about to spring anyway - who cares if the players roll?).
In combat however, you dont use passive perception - hiding becomes a contest, per DMG (ie. the hider takes action/bonus action to hide with active stealth roll vs "free" active perception roll by observers. Not unlike being told a lie and getting a "free" insight check to detect something is odd, or a "free" athletics check vs being grabbed, and so on). Or, at least, that is one interpretation. The hiding rules/passive perception rules are not terribly clear, which if I remember correctly the devs said was by design, to allow DMs to rule in the way their table prefers.
The more I think about passive perception, the more it irks me. Static trap vs static perception = blurgh, as does "easy" hiding due to only one side rolling (and generally woeful passive perception scores across the MM). I think I will go back to just getting the players to record 10 perception & insight checks at the start of every session, and I will select one randomly if/when needed during a session (combined with sometimes making rolls behind your screen for no reason, that is more than enough to "keep 'em guessing").
I dont mind the idea of passive insight so much. I guess because it is not combat related.