Interstate 24 will carry you from Nashville to Chattanooga in about two hours, past Murfreesboro, over the plateau at Monteagle, down into the river valley. It is a well-engineered road and it shows you almost nothing. Every interstate makes that trade and most of them get away with it.
The older way south runs through the Highland Rim on state highways that were the main road before 1960 and have been steadily forgetting it ever since. Through Bedford and Coffee counties the towns still sit on the road rather than beside it, which changes what a town looks like from a car in a way that is difficult to describe and immediately obvious.
You will want to stop in Bell Buckle, and you should, but the more interesting stop is whichever courthouse square you happen to hit around noon, because the plateau counties still keep their business in them.
The Monteagle grade is what people remember either way. The interstate takes it in a long engineered descent with runaway-truck ramps cut into the shoulder. The old road takes it in switchbacks, past overlooks that were there before the highway existed and were built for cars considerably slower than yours.
There is history along this route that the historical markers handle unevenly. The Highlander Folk School operated near Monteagle and trained organizers who went on to shape the civil rights movement, and the state revoked its charter in 1961 and sold off its land. You will not learn this from the road. It is worth knowing while you are on it.
The finished piece gives turn-by-turn directions, the stops that repay the detour, and an honest note about which stretches to skip if the light is going and you still want to make Chattanooga for dinner.