I believe that Curing and Unmolding are the most important parts of the soap making process. These are the times where you'll know if your soap is a YAY or NAY. Soapmakers (the beginners, most oftenly) always look beyond these parts.
What's the difference?
Unmolding is the first 24 hours of removing the soap from the mold.
Curing js when you "cure" your soap after unmolding. Usually take 2-4 weeks.
Beginners will usually want to remove the soap from the mold upon the first touch of the finger feels that the batter is "hard enough". I tell you from experience, DO NOT. Always be patient and let that soap harden and cure. Let it sit overnight and viola, remove it from the mold but don't use it yet. LET IT CURE. You'll see what you might've expected or not from this point in aesthetic issues.
The problem with removing the soap from the mold in less than 24 hours is that you might get the shape you would not have wanted. It might be opposite the shape of the mold.
Here's an example of what I meant when I said that always let the soap harden for 24 hours. I removed the soap from the mold after 2 hours when I thought it was hard enough to be out of the mold. When i came back to my workplace, the soap was flat rather than barred in the rectangular shape. Lesson learnt: BE PATIENT.
After you have unmolded, this is the part when you let caustic parts of the soap settle. You wouldn't want to be using a soap that is caustic! That would irritate your skin. So, curing is vital to the usage of your soap.
Always remember that only cold process soaps require curing. Hot process soaps only require a small span of time for "drying" because the saponification of HP has already been completed unlike CP.