Personally, the Shiratani trail is more beautiful than the Jomon Sugi trail. If you have time, do both. Else, Shiratani is an easier and nicer trail, with many big cedar trees as well.
Good thing about Yakushima trail is that you don't need good weather as most of the trail are within the forest with little opportunity to view the landscape of nearby mountains (except Toikawa Rock). If fact, the moss gets more vibrant after a rain.
We did the Bugyosugi Cedar Course plus Toikawa Rock loop, which should take 4 hours, but we spend the entire day in Shiratani (from 8am) and left on the last bus at 4.10pm.