Oh, gosh, lots of things ~ base oils used, whether it's made with fragrance (synthetic) oils or plant essences (essential oil), additives, what mass manufactured soaps DON'T have (like glycerin), super fatting oils (shea butter, cocoa butter, hemp seed oil, macadamia, etc. . .), how the soap is made.
My favorite soap used to be Caress ~ I can't stand the stuff now, but back in the day, when it seemed expensive and luxurious to use, I'd buy up as much as I could when it went on sale. I loved the way it made my skin SMELL, but it did dry out my skin and I'd have to use lotion to fix the problem. A lot of mass manufactured soaps use beef tallow as the main base oil because it makes a nice, white, hard, long-lasting bar of soap and it's cheap to obtain, then they extract all the naturally occuring glycerin and put it into the lotions they make ~ they sort of get you twice here. They take out one of the important components in soap that makes it moisturizing, thereby drying out your skin, and put it into lotion that you buy to rehydrate your skin.
Anyway, there are lots of reasons why one soap is better than another. The best reason to choose a soap is because you like it best, regardless of what's in it.
J