Actually, all soaps are detergents, because “detergent” is a broader term than “soap”. One chemistry book author proposed the term “syndet” to distinguish soaps from synthetic detergents, but it did not catch very much attention, since it was a rather silly term. As the previous author indicated, the key to detergent activity is for the detergent molecule to have an ionic “head”, almost always small in comparison with the rest of the molecule, and a nonionic “tail”, making up the bulk of the molecule.
The explanation of how detergents work was not very deep in the original article, so please allow me to expound on it. In a water dispersion, detergents (including soluble soaps) form “micelles”, which are little bubbles of the nonionic bits associated with each other in the middle, with the ionic parts on the outside, surrounded by polar water molecules. Much of the time “dirt” on the hands and body are rich in fats, and those are transported to the center of the micelles, making them a bit bigger. There is a limit to that, so if very greasy “dirt” has to be cleaned, more detergent is required, since it is essential that the micelles have that ionic shell uniformly on the outside so that it stays suspended in the water.
“Hard” water kills soap by replacing the sodium or potassium with calcium, magnesium, and/or iron. Only elements in the first column make soluble soaps, including lithium (too expensive for bath use), sodium (the most common one), potassium (it tends to make soft soap, but our ancestors used it until sodium hydroxide (“lye”) became available, rubidium and cesium (both too rare and expensive) or francium (there is not even a kilogram of that intensely radioactive element in the earth’s crust). Calcium, magnesium, and iron are the most common elements that form insoluble soaps, so they do not have any cleaning power. Hard water is rich in one or more of these ions, and they precipitate the fatty acid moiety into a curd that falls to the bottom of the bathtub or shower, or forms the dreaded “bathtub ring”. Here in the Bluegrass the water is extremely hard with calcium mainly, and once a bar of soap gets a little over half used it is hard to make a lather with it because of the calcium soap. In hard water areas, synthetic detergents are more suited since most of them are not affected by calcium, magnesium, or iron.
The reason that a soapy wash cloth stops working even in soft water is that your sweat contains quite a bit of calcium, and forms the insoluble calcium soap as you wash. If your water is very “soft” (little or no calcium, magnesium, or iron ions in it), you will feel very slick after bathing, and it will take quite a while to wash off the soap. With hard water like we have here, the challenge is to keep enough soluble soap on your body to wash off the dirt, and the soap rinses off in an instant. Well, actually it just precipitates into insoluble calcium soap, here, and does not feel slick, but it is there.
Almost exclusively, laundry detergents are synthetics that are not much affected by calcium, magnesium, or iron. I can take a washcloth after laundering it and it is soft and will take up water immediately. That same washcloth, after going through the shower with me, using soap rather than synthetic detergents, dries out hard, coated with insoluble calcium soap. To wet it after it dries requires warm water and physical squeezing to break the barrier of calcium soap coating it. Just running cool water over it does nothing except to waste water, as the insoluble calcium soap is akin to the oil on a duck’s feathers.
Well, there are the technical reasons about how detergents work and why soap fails in “hard” water. Most “body washes” are synthetic detergents and so do not suffer from that problem. The way to tell whether you have soap or a synthetic detergent is to read the ingredient statement. Soap is most often sodium tallowate (lye and beef fat), sodium cocoate (lye and coconut oil), or sodium palmkernelate (lye and palm kernel oil), and there are a few others. On the other hand, if it says sodium lauryl sulfate or sodium lareth sulfate, it is a synthetic detergent and is more immune to hard water. Sometimes synthetics are called anionic, cationic, or nonionic surfactants, and these are also more resistant to hard water than true soaps.