Bar soap.  That humble lathery goodness so many of us grew up with.  For years, drowned in the bath tub by children of all ages, mocked by marketers as they flogged their detergent based alternatives, and the punch line of many a good joke.  Bar soap has a bad reputation, and it is not entirely undeserved.  
Mass-produced, or milled/triple-milled, soap bars are generally poor performers when it comes to skin care.  They are manufactured by a system intent on maximising profit and shelf life, and all other considerations are ignored.  While soap can be full of great skincare properties the common milled soap bar is anything but.  
Fortunately, we don't make milled soap.  But to appreciate the difference we need to understand the processes.  Soap is created when fats and oils (triglycerides) are combined with sodium hydroxide and water. The chemical reaction is called saponification and results in soap + glycerin, and this is where our processes diverge.
How milled soap is made:

During mass-production the soap mixture is boiled to speed the reaction and then the hot soap batter is combined with salt. The salt forces the pure soap to clump together and rise up to the top of the vat where it is skimmed off. They now have 2 products; pure soap, and salty water.  The skimmed soap is washed, dried, bagged and sold to finishing plants around the world, and the salty water is refined to recover the glycerin and unsaponifiables for use in other products.

Later, at the finishing plant, the imported soap noodles are mixed with up to 4% extra ingredients, including fragrance, colours and sometimes milk or extra oils, then milled.  The resulting bar is the skincare equivalent of plain white bread.

What about Cold Process Soap?
During the cold process the soap mixture is mixed with colours, fragrances and any extra ingredients like goat milk, and then poured into moulds and allowed to saponify over many hours or days.  After that it is unmoulded, cut and allowed to cure for as long as it takes to create a hard wearing, mild bar.  The extra ingredients are all still there in the final bar.  It’s the skincare equivalent of fresh, wholegrain, sourdough bread!
So what are these extra ingredients that remain?
Glycerin - Glycerin is a humectant that draws moisture from the surrounding environment and traps it. Glycerin is not added to soap, it is released from the triglycerides during the soap making process. 
Glycerin works to hydrate your skin in the same way as hyaluronic acid (HLA); by trapping moisture within the outer levels of the skin, helping to improve water retention and fight dry and irritated skin. Glycerin molecules are smaller than HLA which means they can penetrate a little further and work their magic from deeper within. You will find glycerin in many lotions and moisturisers
A cold process soap contains about 9% glycerin by weight, but is less than 0.5% in a mass produced bar
Extra Oils - The cold process method generally involves the addition of extra fats and oils. They work to lock in moisture, soften and protect skin from the elements
Fats and oils are essential to healthy skin as they work to trap moisture close to the skin and protect against the elements. Working alongside glycerin the extra oils in our soap combine with the water to create a lotion to soften skin and protect against water loss.
Unsaponifiables - Unsaponifiables are the components of fats and oils that don’t create soap. This is an umbrella term for vitamins, minerals, antioxidants and so on, either from the fats and oils, or from additives like goat milk
When it comes to unsaponifiables cold process soap is king. Due to the way mass-produced soap is made any unsaponifiables are simply washed away. Your skin can benefit from these unsaponifiables, saving you the extra cost, and effort, of using lotions and creams.
So does that mean that mass produced soap is bad?
No. If the goal is to get clean then regular milled soap will get the job done. If you want to help your skin to stay moisturised and healthy then you can choose cold-process soap for the extra glycerin, oils and unsaponifiables.
And all cold process soap has these things you mentioned?
For the most part yes, but in differing amounts. Glycerin, the true skincare powerhouse of cold process soap, will be similar independent of who makes it. But when it comes to extra oils and unsaponifiables then things can vary.   Skills and knowledge will play a part, as will access to ingredients and the soapmakers personal interests. You can make soap from premium fresh ingredients and you can also make it from rancid ex-cooking oils.  One is a great soap full of antioxidants, and the other is helping to clean up the planet by keeping used oils from landfill.  No one soap is wrong on all counts.  But not every soap is right for you and your needs.