The small knife is used for a number of handicraft task, I think, and is rather essential in Buiodda's Crafts mod.
The hunting knife is a fighting knife, essentially, and can be seen as a generic alternative to the Northern or the Kaumolais knife.
The Fisher's knife isn't preferred for anything in vanilla, as far as I know. I guess it would be good for cleaning fish, but there's no such task in UrW currently.
The "knife"(i.e. without any specific purpose) isn't preferred for anything, unsurprisingly.
The splitting axe is also used to make fire wood (which you don't really need in vanilla).
As far as I understand, using a non preferred tool counts as using a preferred one of one quality level less. You're informed that your tools are to blame, not your skill when you would have succeeded had the tools been up to the task.
There was recently a thread where quality effects was touched upon, and that indicated you received a success roll bonus from masterworks quality tools, but nothing from fine quality (while fine and masterworks gave increasing bonuses for fighting). However, I can't say I've noticed any increase in the number Superior skins gained with a masterworks broad knife compared to the fine one when I finally was able to upgrade.
It can also be noted that the change log quote provided by Privateer is somewhat misleading, as you can't make wooden bowls with a quality better than decent anymore, because the resulting item quality is capped by the quality of the input item, and wooden blocks are all decent. The item quality logic still applies, though, and you can still produce poor and leaky bowls.