A question for the philosoraptors :v
For the same reason a salt-water solution in a closed half-filled bottle will not evaporate - even though it is in contact with some air.
Open the lid - and eventually the water evaporates to leave just salt.
The slime solution is in a "balance" with the air in the tube - since it is a "closed" system. When the tube is punctured the slime flowing out of the puncture is no longer in a "closed" system and evaporates / dries in the open air.
Takes me back to Chemistry and reaction kinetics (yuck) .. but anyway that's my take on it.
According to the name-brand Slime supplier's own article on The Science Behind Slime Tire Sealant :
Slime repairs flats with a mechanical seal, meaning physical particles actually plug up the hole. There is no chemical reaction at the puncture site.
These physical particles are a combination of long and short fibers, as well as rubber particles (Those little black bits you see mixed in with the green). We call this special mixture of particles Fibro-Seal technology.
So the sealant's liquid is just a water-based carrier to help the particles move to where they are needed. It doesn't dry out (much) in your bicycle's tube for the same reason it doesn't dry out in the bottle, just because there's not really any place for the water to go once the air in the tube gets to 100% humidity. And based on the Slime company's own description, it never really needs to "harden" so much as it just ends up clogging up any holes.
Hypothetically one might even be able to re-hydrate old Slime if it dries out (unless there's other volatile parts of the mixture that go missing too); the manufacture does not provide instructions for this and would probably rather you buy more instead!
Think more "leaves in a gutter downspout" instead of like any sort of glue or epoxy. Without water the leaves just blow around, but when rain rushes along the gutter the downspout can get "sealed off" by the junk it brings with it.