I haven’t gone both ways, but when I climbed those mountains I started at the Imp Face parking area and climbed to the ridge just south of North Carter, then followed the Appalachian Trail south over Middle and South Carters, Mount Hight, and Carter Dome. I continued down to the Carter Notch hut and descended Nineteen Mile Brook Trail to Route 16 then walked back to my car.
I think it was about 16 miles all told, maybe about ten hours or so, I can’t remember the details now. As far as advantages go, you could check the AMC guide and see if there’s elevation gain listed for both directions, and if there is then see if it’s markedly more one way or the other. I guess if you begin and end at the same spot as I did, it would be the same gain either way. Spotting a car someplace, you’d have to figure out which way is easier, if one way is.
It was steep going down into Carter Notch, but I don’t know that it was so steep that you’d want to go up that way instead. It was a long grind getting up to the ridge past Imp Face, too.
Be sure you don’t pass by Mount Hight, or miss the trail from its summit (sharp right turn, southbound) to Carter Dome.