What does planting woods with your worker give you? I understand old forests give you appeal but it doesn't tell me what planting new forests even does for me... when where why should I plant them?
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.
You can increase the appeal of neighboring tiles. This is good for seaside resorts (more gold and tourism). It is also useful for creating or boosting the tourism bonus from national parks. There is no difference between an old forest and a new forest (except that you don't get any production from chopping a new forest).