All skills will decrease stamina usage. The higher skill level, the less stamina is used for using that skill. Weapon skill levels will increase dmg, but you wont feel the difference that much as in stamina usage.
The best skill where you can tell the difference is jumping skill, because you will be able to jump higher and further.
I have been testing spears in debugmode. Not sure what were exact numbers, but if you use 10 stamina for one hit with a spear on skill level zero, then you will use 5 stamina at skill level 100. So its basically one stamina point less used per 20 skill levels.