I've never heard of a "trail" around Cade's Cove... I thought the entire GSMNP was shut down to OHV. Please post more if you find anything out.
Are you talking about Parson's Branch Road? It has a sign saying something like, "4WD or High Clearance Vehicles Only", and it's closed during the winter, but it's just a narrow gravel road. It's one-way out of the park, between Cade's Cove and the Dragon (Hwy 129). It has a few steep hills that front-wheel drive vehicles and maybe light pickups have problems climbing, and low-clearance passenger cars will drag in spots. It also has several wet creek crossings with maybe a few inches of water over the concrete swale. Generally any decent clearance vehicle can make it from a Subaru Forrester on up. I drove it a few years ago and never got out of 2Hi, unless maybe I put it in 4Hi climbing some of the steep hills.
The biggest problem with Parson's Branch is that it's only accessible from the Cove itself. After mid-morning during the summer the Cove is a non-stop multi-mile traffic jam.
Or maybe Rich Mt. Road. I haven't driven that one, but I understand it's similar to Parson's Branch. It runs one-way north from the Cove to Townsend. Looking at topography it looks more interesting than Parson's Branch, and much shorter. When we were up there we needed to go west so we picked Parson's.