Is there any difference here at all?
To add more nuance (and confirm what's already been said) to the difference, I find (in AmE):
- you can say 'a forest' (in general), or 'the forest' (a particular one), and 'the woods' (the one that you're walking through), but it would seem strange to refer to an collection of trees as 'a woods' (because of number agreement) or 'a wood' (because that would refer to a particular piece of lumber.
- 'forests' are bigger than 'woods'. But a small collection of trees is not necessarily 'the woods'. Something smaller (without specifying actual size) would be called 'a stand of trees'. (i.e. if you can go into the collection and not see the end of the trees, then you're not in a stand of trees)
- 'forest' has a slightly more official, formal feel to it than 'woods'.
It normally comes down to scale. I wouldn't consider the half-acre of trees behind my house a forest, but I frequently describe it as "the woods out back".
On the flip side, while I would typically describe, let's say a national park as a forest, I might also say that someone lost there is "lost in the woods"
These are almost completely synonymous, but I'd say a forest is a more defined (and probably larger) set of trees than woods. As in, if it has a name, it's almost certainly a forest. If it's just a clump of trees that may or may not end after ten feet, it's woods; if, however, you know that it stretches for five miles in each direction, then it's a forest.
Originally, 'forest' meant a royal hunting ground, which is why they are usually larger than 'woods'; woods can be just a few trees, whereas a forest is usually much larger and denser, both in trees and vegetation. Also in some places such as the UK, woods can be plural because each 'wood' may be in some way separated or distinct.
Forestry is coniferous trees planted as a crop to be harvested. It doesn't matter how small or large the stand of trees is, it's still forestry. I suspect that has influenced my usage, so that to me, a forest may be coniferous or deciduous, but my initial image would be coniferous. By contrast, a woodland is always deciduous, established, broadleaf trees, widely spaced but with little undergrowth, easy to walk through.
I wouldn't usually say a wood. I do go for a walk in the woods. These too are always deciduous in my usage, but I wouldn't be confused by another person using these words for coniferous trees. Perhaps all this is influenced by the fact that there are very few natural stands of conifers in Ireland. When I was in Colorado, the pine forests hugging the hills felt quite dark and oppressive, and the lack of any broadleaf other than scrub oak began to get to me.