In sum, what advantage does owning and reading Volo’s Guide to Monsters in-game give a PC over those that do not?
Depends what's in it, depends how accurate what's in it is and depends on how much the PC can recall. Further, there is how you deal with any distinction between the knowledge of the PCs and the players.
What's in it
Volo's Guide to Monsters is (obviously) two different books - the real one published by WotC and the in-universe one written by the titular Volo.
As the DM, you need to decide if they cover the same or different subject matter - it doesn't necessarily follow that what is covered in the real book is what is covered in the imaginary book.
How accurate is it
Volo "was a legendary traveler and storyteller" - it may come as a surprise to you, but not everything written down by "storytellers" (or, as they are known in our world, journalists) is factually accurate in every respect.
Sources of error include:
- Errors in primary research: rigorous scientific research of "monsters" is naturally hazardous and some fudging may have occurred.
- Secondary source: Partly for the above reason, much of the work is unlikely to be based on first-hand experience. So you have Volo imperfectly reporting what someone imperfectly remembered from an encounter where they were probably in an emotional state that did not allow for unbiased, objective observation and analysis. This is why Pliny's Natural History contains such gems as the Cynocephalus, the Monopod and the Catoblepas.
- Conflict between ethical and commercial imperatives: Volo's primary objective is to sell books. His secondary objective is to maintain a reputation for accuracy and honesty in order to sell more books. It is only his tertiary objective to actually be accurate and honest.
- Typos: As exemplified by Douglas Adams in The Hitchhiker's Guide to the Galaxy:
This was the gist of the notice. It said "The Guide is definitive. Reality is frequently inaccurate."
This has led to some interesting consequences. For instance, when the Editors of the Guide were sued by the families of those who had died as a result of taking the entry on the planet Tralal literally (it said "Ravenous Bugblatter Beasts often make a very good meal for visiting tourists" instead of "Ravenous Bugblatter Beasts often make a very good meal of visiting tourists"), they claimed that the first version of the sentence was the more aesthetically pleasing, summoned a qualified poet to testify under oath that beauty was truth, truth beauty and hoped thereby to prove that the guilty party in this case was Life itself for failing to be either beautiful or true. The judges concurred, and in a moving speech held that Life itself was in contempt of court, and duly confiscated it from all those there present before going off to enjoy a pleasant evening's ultragolf.
Notwithstanding, the PCs have spent good money on this and it should be more useful than not as a result.
Circumstances will dictate if they need to remember what they have read (assuming they have read it) or if they have the leisure to look it up.
If they have to remember it, I follow a very simple rule: player skill trumps character skill - if the player remembers then their PC remembers if the player wants (they may not want for role-playing reasons but that's up to the player). If the player doesn't remember then the PC might remember based on a feat like Keen Mind or an ability check.
Tell your players the stats: there's no harm in it.
The stats are the player's way of understanding the threat the monsters represent in-game. They know their own stats and that allows them to understand what their PC can do - knowing the monster's stats is no different. In any case, they would know this stuff after their first combat encounter anyway.