In short...It's the other park that Dolly Parton didn't buy..
Yes, the two parks are affiliated with each other. Herschend Family Entertainment is the company that has interest in both, and both parks were known as Silver Dollar City at some point (though the park now known as Dollywood was known by at least two additional names before it was ever called Silver Dollar City). Dollywood came about because Dolly Parton wanted to give back to the area she came from in the 80s (she is from Sevierville, just a town over from Dollywood, which is in Pigeon Forge, TN) by building a theme park. The Tennessee property was already struggling at that point, and the Herschends knew they couldn't compete with her, so they decided to see if she'd be interested in a partnership. She agreed, and the rest is history as far as that was concerned.
That said, dare I say I actually prefer Silver Dollar City to Dollywood? There are lots of little touches that don't (or no longer) exist at Dollywood, and of course they have the cave tour as well. Dolly Parton undoubtedly saved the TN park and brought with it a fair amount of attendance and publicity, but there's something of an identity struggle at the TN park because of it IMO. SDC is successful in its own right, but more modestly so due to a combination of location and not having a prominent national backer. Though there are better parks strictly for coasters, I'd say it's the best proper theme park in the Midwest. The rest of Branson doesn't really do much for me, but I always try to make it out to SDC a few times a year.