So what's the difference?
- sodium hydroxide (often called just "lye") makes bar soap - solid, opaque bar soap
- potassium hydroxide (often called "potash") makes liquid soap - flowing, clear or translucent liquid soap
Old fashioned or "pioneer" soaps were made from lye that was made with wood ashes - and were primarily a soft, gooey, soft soap. Wood ashes produce mostly potassium hydroxide. I've seen old instructions/recipes that said "add in a handful of salt until the soap thickens" - that appears to be just adding some sodium to the mix to make it firmer.
Luckily, today we can get both sodium hydroxide and potassium hydroxide in pure versions with consistent strength from chemical vendors. Unlike Grandma's unpredictable "lye soap" - we can know that our soaps, whether liquid or bar, made with sodium or potassium hydroxide, are going to be gentle on our skin.