If you have a hex with a hill and forest, you get one food and 2 production.
If I then but a lumbermill on it, the production get's increased by 2, but if I remove the forest and then build a mine on it the production also get's increased by 2. (First you lose one but then get 3 production)
Q: Why should I ever build a Lumbermill, when I have the same outcome with a mine but still get the production-bonus from cutting down the forest?
To clearify:
You get more than 50 production for cutting down the forest and you just need one charge of you builder.
EDIT:
I didn't express myself fully, the only Condition I want to be met is, that it's forst + hill. But if there are other boni I didn't notice (e.g. Tech that increases lumbermill production) please hint me on them.
Best Answer
If you consider only the points you specified, the answer is simple. Just build the mine :).
There are only 2 reasons you should be considering while doing this (hills with forest - bear in mind that lumbermill can also be constructed on tiles without hills but with forest):
Religion that gives bonuses on forest tiles (I'm speaking from memory but afaik there was one bonus like that)
Buildings that require adjected forest (I think there was a wonder that require that... same as above, I speak from memory but I think there was one)
District adjacency bonuses like +1 faith for every two adjacent forest (woods) tiles
And... that's kind of it. This is only considering your exact scenario.
For endgame scenarios, with expanded cities you should read this post: http://forums.civfanatics.com/threads/mines-or-lumbermill.391185/ - apparently in endgame lumbers become better than the mines (it is a civ 5 post but I think it still applies here as well, although a bit less since districts and placed wonders limit the possible space).