While both are wonderful and storied resorts, the big difference is development scale.
The Greenbrier has morphed from resort into a sprawling residential community of on-site dwelling clusters, linked by constant paved roads everywhere, all gated and walled off from the fast food and gas station town of White Sulphur Springs, into which guests will rarely venture other than to enter and exit the resort through the Greenbrier's security-restricted passages. In general, the feeling is busy, crowded and occasionally overwhelming, consistent with the interior decor of the Greenbrier which is loud and riotously colorful.
The Homestead is the opposite, with almost no pavement or residential clusters within sight of the resort, leaving the ground's landscape natural and accessible only by foot (or golf cart or bicycle) and creating an entirely different feel. Rather than being walled off from the resort, the village of Hot Springs has been thoughtfully preserved, upgraded with light retail and dining, and integrated into the resort as a natural extension of the setting. There are no security checks, walls, gates or other barriers to define the resort's boundaries, which merge seamlessly into the natural setting or the village, consistent with the subtlety of the Homestead's subdued color schemes indoors.
The two legendary venues are only an hour apart so if you stay at one, visiting the other is a worthwhile sidetrip.