It's difficult to explain properly, but yes, progressing upwards thru the finer stones will make your knife cut better. It's just that the primary sharpening has to happen with the coarser stones in order to get the benefits. The first stones will cut the primary bevel- if you choose to microbevel you can do so but even that can be done with a 1000 grit or so. Finer stones will refine the edge to a great deal, but only if the edge is there to begin with.
So yes, in practical terms you do use finer stones to get your knife "sharper" in a sense, but I still don't really like to say it that way. A polished edge will definitely do things you can't do as well with a "rougher" edge, like push cutting and slicing tissue. As already stated, you can easily see that you're doing less damage to the food by finely julienning basil or slicing tomatoes very thin. The polished edge glides thru food with very little effort or pressure vs a less polished edge. Still, many feel that a "toothier" edge is better for stuff like cutting cardboard and leather.
please tell me where those places are, so I can make sure not to eat there