When you're out driving around
If you're not on a road, you're probably on private property unless otherwise indicated (at least for places East of the Rocky Mountains... the situation in the West and Alaska is quite different.) National or state parks would be the primary exception to that. Granted, there are many of those in Tennessee and North Carolina.
If there's a specific property you want to visit and you know it is private, you really should seek permission from the owner first, unless the owners have indicated that their property is open to the public.
The lack of a fence or "No Trespassing" sign does not mean that the property isn't private or that its owners welcome the public to wander around their property.
When you're on the road, public roads will usually have street signs. Roads with no signs at all are more likely driveways which are privately owned. A mailbox and/or sign indicating a street address beside the road is a good indicator that it's actually a private driveway and not a public road.
Regarding the particular Google Maps link in the question, unfortunately, there's no Street View there, so it's not easy to tell if there are any signs there.
Geographic Information Systems
When you're looking up properties online, fortunately, GIS is a thing. It turns out that some (all?) U.S. states have public GIS systems with data from the county property ownership records. North Carolina and Tennessee are not exceptions to this. Unfortunately, I don't immediately see a way to provide a link with a particular place focused in the North Carolina system's interactive map. However, I was able to find the particular property you linked and it is indeed a single private property.
The area outlined in teal below is the property on which the road you pinned lies, so it is indeed just a private driveway, as it appeared. The actual road you pinned is not shown on the map below, as it's not a public road, but you can see the intersection of the public roads by which it lies.
Screenshot of pinned map location on NC's Parcel Map
Tennessee has a similar system.