When I first started making soap I used the cold process method. After mixing the oils and lye solution you pour the soap into a mold of some sort and wait for four or more weeks before you can use the soap. Saponification doesn't usually take quite that long but you want to be sure there is no live lye in your soap and you want to give the bars a chance to harden.
With hot process soap you basically cook the soap batter either by heating it in a crockpot until it volcanoes and goes through applesauce, mashed potato and Vaseline stages, or by soaping at high temperatures (in the 200F area) in a stainless steel pot/plastic container, all of the stages happen pretty quick when soaping that hot. It's pretty fun to watch this all take place within a half hour.
There are a few advantages to the hot process method. First, it's fast, once the cook is over, saponification has taken place and you can use the soap immediately. Usually you give it a few weeks to evaporate some of the moisture and give the bar a chance to harden. Another advantage is you're able to add the superfats to the soap after saponification so you can add the really good fats then, giving you the chance to control which awesome oils aren't eaten up by the lye. It's good practice to use the more pricey oils at this point. It can also help to add essential oils after the cook so they're not superheating and loosing their oomph.
There are a couple of disadvantages to the hot process method as well...at least for me. One is, the soap batter is pretty thick and while it's possible for some soapers to get some amazing swirls, it's not as easy to get the pretty bars that you get with the cold process method...at least for me. Another disadvantage is that there can be quite a bit of stirring of the hot batter which is like stirring a mix between brownie batter and cookie dough.
Due to the job I have, my hands and wrists already take a beating and stirring a thick batter gets extremely painful. Over the summer I've been working on cold process soaps because it's not as hot as working over very hot soap batters. Making the cold process soaps made me realize how much easier it was on my hands and wrists and I'm thinking I may have to go back to that method.
One concern I have is not having the ability to choose which oils/fats will superfat my soaps. I'm not even sure it will be noticeable. I'm going to give it a try and see if there is any difference in the properties of the soaps. Quality, moisturizing soaps is my number one goal. I'll do what works best to achieve that!