Most of the time, the generative dimension is something you only experience indirectly.
4D Golf doesn't use as much procedural generation as you might think, but it (and other 4D games) gives us a good illustration of what interacting with a possibility space is like. For any given world, we become acutely conscious that it could have been otherwise.
Sometimes, of course, we prefer it if the generated artifact stands on its own, downplaying the might-have-been nature of the process that created it. But, just as real-world objects gain part of their aura from their history of creation and existance, generative artifacts retain a penumbra of possibilities. It means more that this is red when it could have been yellow.
This is difficult to directly experience (unless you obsessively play Mu Cartographer, I suppose). A game that intentionally lets you access the possibility space directly can give you a glimpse into that larger world.
It can get a bit distracting, of course. And if you try to build one, you're going to become acutely aware that which direction you can move in generative space matters quite a lot for the experience.
If you work with generativity, though, I think it's worth it to play around with something that lets you tweak the levers in real time, so you can get a feel for the decisions about decisions that your decisions are shaping. I think my earliest experience of this was playing around with Perlin noise and seeing how the shapes changed as I adjusted each parameter, sliding around the generative space as I sought out the region that felt right.
There are other ways to incorporate the patina of generativity into the experience of the artifact, of course. But I happen to like how direct this one is.