Ok, so here's the deal.
I think Elves should definetly be more okay with wood-cutting, but create artefacts and items that more "naturally authentic". To put it simply, whereas dwarves create complex items with various details made of different metals and materials, mostly stones, metals and gems, elves create wood items that look like strong branches that were bent into that shape.
For examples; an elven bow would be curvier, have a very rustic and authentic look to it, and mostly be composed of non-industrialized component, such as, let's say, spider webs for the bowstring. It could be decorated with leaves, sea shells, gems or tree amber, or any other surface natural ressource. Now you may say that they're already doing this, but I suggest that Elves should be able to grow exotic types of trees and maybe expand a bit more on the items they create.
Their villages, settlements and cities could be mostly made of rocks and blend in with nature, rather than having them literally living in trees. For example, they can carve mountains into castles and let trees grow all around, while planting flowers to symbolize different places. Maybe implementing flower language could be a thing?
Another interesting thing would be to have elves master different crafts that dwarves don't usually practice. Maybe have them being able to make more glass items and give to an elite of elves the ability to bend wood into various shapes to create items (wood bending)?
Another interesting thing that could be implemented would be the natural order among elves. Since they are closer to nature, the thought of having Elves who are naturally superior or more adapted to a middle than the others could be an interesting take on their closeness to the ecosystem. As Betaking suggested, High Elves could come into play. They would basically transcend the elves by being able to manipulate plants, trees and crops, or be yet closer to magic. Dark Elves could be their polar opposite, completely alienated from nature and refusing to embrace it (they could maybe be the fruit of the clowns, giving the elves some rivals?). Drows would probably be the weirdest, I guess. Being both friends of nature, but also very fond of industry. Kinda like them giant megacorporations selling eco friendly products, but taken to another extreme.
Also, having elves that correspond the biome they're in could be interesting; so like, if an elf was born in a cold area, he would feel lost or unfamiliar with hotter biomes. A snow elf in the desert would lose his mind over how the sun completely dries them. Deep and Hill dwarves already have some minor differences, such as being more likely to be sun-sick if they live underground, but as of now, those differences are quite minimal. Tarn said he would expand on that, so I hope he does the same with elves.