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Horizon Forbidden West will not offer a free upgrade from the PlayStation 4 to PlayStation 5 on either its Standard or Special Editions. You also cannot pay to upgrade from the Standard edition to an edition that does include the PS5 upgrade either.

As it currently stands, PS4 and PS5 will receive totally separate versions of the same game. You will be able to play the PS4 release on the PS5, but it won't have any of the PS5 specific visual and performance upgrades included with it. This lack of an upgrade path was revealed by way of the game's FAQ page.

Can I upgrade my Horizon Forbidden West PS4 version to the PS5 version?
To access both the PS4 and PS5 versions of Horizon Forbidden West, you need to purchase the Digital Deluxe, Collector’s, or Regalla Editions. Dual entitlement does not apply to the Standard and Special Editions.

Can I upgrade my Standard Edition to the Digital Deluxe edition of Horizon Forbidden West?
There are no plans for an update offer from Standard to Digital Deluxe Edition at this time.
The only way you get both the PS4 and PS5 versions of the game with a single purchase is if you buy the more expensive Digital Deluxe, Collector's, or Regalla Edition. Typically, Sony has been pretty stingy when it comes to free upgrades for first-party upgrade titles. At the very least though, they offered fans a cheaper upgrade path if they wanted to jump from the PS4 version to the PS5 version. That is not the case here.

This means that, at a minimum, you will have to spend $79.99 (USD) to get a version of the game that offers dual entitlements to the PS4 and PS5 versions. This is the Digital Deluxe Edition of the game. The Collector's Edition of the game will set you back a whopping $199.99. And the Regalla Edition? It's $259.99.

Also, yes, the PS5 Standard Edition is indeed $10 more than the PS4 Standard Edition.