The main hand and off hand tools have different crafting stats which are likely to have different effects on your outcome. What I found most interesting is "magical craftsmanship", which seems to imply a higher success rate if your stats are more magic-oriented. Indeed, as a THM I have found much more crafting success with my offhand tool, which is more in favor of magical craftsmanship, than my main hand tool. This not only affects the overall success rate of my synths, but also seems to have an impact on starting quality and durability as well.
I don't believe individual recipes favor one tool over the other, rather the effectiveness of your tools are influenced by other factors. At higher levels, I imagine main hand will favor consistent normal quality crafts, while offhand will favor less consistent high quality crafts. Or perhaps vice versa depending on the class. Plenty of experimentation to be done.
But the tool does effect the outcome. http://ffxiv.zam.com/en/item.html?ffxivitem=10005202#Ingredient_in
Notice two quantities for hempen yarn. http://ffxiv.zam.com/en/recipe.html?ffxivrecipe=13
And in the comments, it seems someone has already figured it out. I heard about it in game.
But to be safe, I'll try this myself to verify if it is true. Well, as soon as I get enough lightning shards.