Why don’t the layers diffuse onto eachother?
They do, you just didn't wait long enough. If the densities are different enough, sugar-water solutions can stay separated for weeks or even years. Given enough time though, they would eventually form a uniform solution.
How can they stay separated for so long?
For the same reason that saltwater can remain separate from fresh water, or even hot water and cold water - diffusion is typically a slow process, particularly if you do not stir, or otherwise mix the substances.
Consider this: drop a single drop of food dye into a glass of water. There's no way the entire glass immediately turns into the relevant colour, it takes time for the molecules to move about.
Complete spitball here. But I suspect that the syrup on the bottom, being much denser than just water and also not being part of the vapor cloud above the original solution, settled out very fast, and that's your bottom layer. The top layer, in this hypothesis, is the water that condensed from the steam in the air directly above the hot syrup, sliding down the sides of the container and sitting on top without mixing because 1) its remarkably less dense than the syrup so has a hard time punching through and getting more surface area exposed for interaction with the syrup and 2) its also not being mixed or agitated in any way to help overcome the issue in part 1. And the more you stir, the more homogeneous it becomes.
Until/unless it crystalizes, anyway, at which point it gets watery again as the sugar turns into rock candy.
Edit: never mind me. You meant just putting sugar into water. I'm sure some is dissolving still and probably contributing to the effect you are seeing after stirring, but you should just ignore me probably.
