When should we use landuse=forest rather than natural=wood?

I don't quite understand the whole problem. To me, it seems rather clear and logical that natural=* should describe what natural elements are on the area and landuse=* what the land is being used for by people. I don't see why these should be mutually exclusive.

I would expect landuse=forest to almost always have wood-type natural elements, so it would usually indicate also natural=wood. Likewise, people almost always "use" an area with wood-type elements as a forest (e.g. for recreation), so natural=wood usually also indicates landuse=forest. That's why I consider landuse=forest to be one of the most useless landuse values - except in cases like Hawkeye refers to in his answer:

tagging the area of highly managed forest with landuse=forest and the individual packets of woodlands as natural=wood.

However, this seems to not be the way a lot of the community uses these tags, for whatever reasons. The wiki for natural=wood describes it as

Natural primeval woodland. For forests that are managed by someone, use landuse=forest instead.

And the wiki for landuse=forest also states it is for

Managed forest or woodland plantation

However as some argue, trying to differentate between a "managed" forest and a "natural" forest might be rather silly in most of the western world (maybe most of whole world as well), where pretty much all forest is actually being managed, even though it is kept in a "natural" state. Whether this distiction is also in any way relevant to map is a good question.

As with many other conflicting tag sematics, a good solution might be to switch to using some completely other tagging scheme for the purpose. The wiki for Landcover says:

There is not currently a good tag to describe a landcover of trees as opposed to a landuse of timber production for which landuse=forest is appropriate or natural=wood for primary unmanaged woodland. The tag landcover=trees has been proposed for this purpose.

I will, however, personally keep using natural=wood for any things that I can look at and say "hey, that's a wood", and not add any landuse tags. The natural=* tag is already used similarly in for example natural=water

Futher, in a more philosophical sense the whole distinction between "natural" and "managed" is very problematic. If a forest is owned, but the owner doesn't actually manage it, is it natural or managed? What about a forest that had been managed previously, but is now left as unmanaged - is that now in a natural state? If we manage a forest to keep it healthy, but do it so we can make sure it remains in a healthy natural state to preserve it, is the forest really no longer "natural", or in fact more "natural"? And even further, we are not the only things that live on land which shape the land itself.

Tagging really should concern the information that people have use for. It's a very rare case where there's any relevance between a forest that is kept healthy by managing it, and a forest that is kept unmanaged. Both are forests. Forest is a thing of nature, so natural seems like a good key to describe it. Where the distiction between managed and unmanaged matters, that data should have then have an extra tag ([managed=yes/no][7] perhaps?). Likewise, a forest that is not even designed to look natural (those lines of trees) would also be relevant information (compared to a managed, "natural" forest) so that data should also have an extra tag for it, preferably causing a different rendering too. But that's just my opinion, of course.

In conclusion, your question unfortunately has no established, clear answer.