You will not be able to re-download them, or play any games that you had previously downloaded & installed via Playstation Plus. If you try to download any of the games from your download history, it will not work.
However, if you later decide to renew your subscription, these games will become available for use again. They are even re-downloadable, as long as you added the games to your cart and successfully checked out before their free offer period passed.
Q: How long do I get to keep the free games with PS Plus?
A: Games you get with PS Plus are free to download with a PS Plus membership. As long as you are a PS Plus member, you’ll always have access to any game you’ve added to your library, even if you’ve previously deleted a game from your hard drive.
Q: If a game is no longer available for free with PS Plus, can I still download it for free with my PS Plus membership?
A: If you did not add a game to your library when it was free with PS Plus, you will not be able to secure your free download. However, if you added a game to your library when it was available for free, but it’s no longer on your hard drive, you can visit the PlayStation Store and re-download it anytime, as long as you’re a PS Plus member.
Q: How many times can I download / re-download a game in the Instant
Game Collection?
A: Download a game as many times as you like, even if it’s been deleted from your PS4 hard drive, as long as you’re a PS Plus member.
Q: What happens to the games and add-ons I download for free through a PS Plus membership if I decide to not renew my membership?
A: If you do not renew your PS Plus membership, you won't have access to previous free games downloaded with PS Plus. If you have purchased add-ons for a game you received free with PS Plus, and the add-ons require the free game to play, you will not be able to access the add-ons if you choose not to renew your membership.
Should you decide to re-activate your membership, you can again play all free games downloaded, along with all purchased add-ons for these games. Alternatively, you may also purchase the specific game to access any purchased add-ons for the game.
Best Answer
No, there's no benefit for you to not select and download the games offered to you each month. PS Plus gets you a set number of games each month, decided by Sony--not adding these games to your library when available will offer you nothing except a missed game.
As to why the PS Network doesn't just automatically add those games to your library... they've never said why they don't, and I doubt they ever will, but there's a few obvious business benefits to Sony that they gain by not doing this:
Gauge Customer Interest: If 100% of your users get 100% percent of the games each month, you're limited in your ability to see what's popular. If 92% of your users add the RPG you're offering to their library, but only 80% of the users add the racing game you're offering, you now have valuable information about which genres are currently popular with your customer base, and you can potentially leverage that knowledge into more PS+ sales.
Better Traffic to Market: It's a well-known marketing tactic that just getting customers to walk inside your store increases sales, even if they don't intend to buy anything when they walk in. Forcing users to open the PSN store, where they'll be bombarded with ads for new and on-sale games, in order to get their free stuff, follows this marketing principle. (You can see people discussing this tactic working on them in the /r/PlaystationPlus thread here.)
Future Sales: If the game gets added to the library each month, and you forget to download a game you're interested in one month... well, no worries, you've got a copy in your library! The way it works currently, however, if you miss picking up your games then the PS Network--and the publishers of the games on offer--gets another chance at a future sale for every customer who forgets to download.
People May Not Want Them: It's worth noting that, sometimes, giving customers free things will upset them. See the U2 free iTunes album PR disaster for an example. People can have reasons to not want certain games in their library--imagine a religious person not wanting a game about evil gods, or a person with young children not wanting a violent game--and while of course the rational solution would then be 'don't sign up for a service that gives you free games', customers aren't always rational. Sony may benefit from heading similar PR disasters to Apple's off at the pass.
TL;DR the customer gains nothing from not claiming the free games. However, the businesses involved gain several benefits from keeping'not claiming the free games' an option.