In an interview with the Daily Star (warning: NWFW images), Hello Games director Sean Murray said that putting No Man's Sky on beefier hardware like the PS4 Neo would enable his team to do more than merely upgrade the game's visuals.
"If you play another game and if they do a remaster of that game, what they will do is up-res the textures and things like that. That's because the base geometry, the base shape of the world, the way of playing is totally like lost, basically. Hundreds of people have worked on it thousands of hours to create what you see."
Murray provided further clarification. "For our game, it's procedurally generated. So more powerful hardware doesn't just mean upgraded textures or a higher framerate. It means we can fundamentally change the experience."
As the interview progressed, Murray explained that a more powerful platform would let No Man's Sky's designers add more trees, and more leaves to trees, as well as increase world density—details that would serve to immerse you in the game deeper than ever before.
This is all hypothetical, of course, since no one from Hello Games has announced plans to bring No Man's Sky to PS4 Neo (rumored to be unveiled at a Sony press event early in September). But there's another point to consider.
Reports concerning plans for Neo have indicated that Sony wants publishers to keep the playing field even for all consumers. For instance, publishers will not be permitted to release two versions of a game: one for Neo, and another for vanilla PS4. That rule extends to game features; if developers implement a split-screen multiplayer mode in their game, it must be present on both PS4 models.
I don't know if that mandate extends to superficial details like the number of leaves on a tree. Assuming it does, Hello Games would not be able to "fundamentally change the experience" (via Daily Star) of No Man's Sky between PS4 and PS4 Neo, unless they released the game under a completely different title, such as No Man's Sky Remastered or the like.