What are the benefits of planting woods with a builder?
Forest adds +1 production to whatever tile you plant it on. Pretty useful for all sorts of flat land tiles: plains, tundra, grassland. Especially so if you add a lumber mill on top.
In addition to the production bonus mentioned by vanao veneri, note that Teddy Roosevelt has the Environmentalist agenda in foreign policy:
Builds National Parks, doesn't clear features, plants forests. Likes civilizations that plant forests or found National Parks. Dislikes civilizations that clear features.