Vanilla and Rise and Fall
Any Woods feature along a river (especially if on Hills) can become invaluable for industry due to the high Production boost from Lumber Mills. Avoid cutting down Woods in the early game, because being able to improve them with Lumber Mills will provide you with a great boost to Production. What's more, later you will be able to claim diplomatic bonuses for not having cut these Woods, in addition to gaining Appeal bonuses for neighboring tiles!
In Gathering Storm, after the June 2019 Update, the Lumber Mill has been redesigned and enhanced considerably: it is unlocked one era earlier and has +2 Production from the start (which replaces the River adjacency bonus). This makes it a vital Production booster for empires starting in level terrain without Hills, where Woods are the only useful feature. Furthermore, come the Renaissance Era and the discovery of Mercantilism, the Lumber Mill can now be built on Rainforest tiles as well! All this makes the Lumber Mill a viable alternative to the Mine as a general Production booster for areas where Woods and Rainforests dominate. It also has the advantage of not requiring the Woods or Rainforest to be cleared from its tile as a Mine does (unless the tile also contains a resource), which has more implications than just the diminished Appeal in Gathering Storm.
For centuries the purpose of a lumber mill has not changed: logs go in one end, and cut beams, boards, posts, and shingles come out the other. The earliest known sawmill that can be reliably dated is a Roman water-powered mill uncovered at Hierapolis dating to the 3rd Century AD. Small, water-powered mills were common throughout Europe and the Middle East where trees were plentiful, and the designs later went with European colonists so they could start chopping down the virgin forests everywhere else as well. The Industrial Revolution sped the whole process, as steam railroads could transport the logs to steam lumber mills, and then haul the “finished” (in more ways than one) wood products to market in the booming cities. In time, even the byproducts of the trimming and cutting were used – for paper, fuel, and packing.