Reddit user amaranth1977 made a great post about soap:

Most modern liquid "soaps" are not technically soaps at all, which is to say they're not produced from mixing vegetable or animal fats with a strongly alkaline solution such as lye or potash. Instead they're a blend of (usually) petroleum-derived surfactants such as sodium lauryl sulfate with other chemicals to produce a detergent that matches the desired use..

Shampoo is designed to be gentle on the keratin which forms hair, have strong foaming properties to be more easily worked through the fine strands, remove common hairstyling products, and - especially for those of us with more than a couple inches of hair - have specific effects on the texture of the hair. It has a fairly low concentration of surfactants so that it rinses out quickly and you're not in the shower forever trying to get it all out of your hair.

Hand and body wash is usually formulated with a mild surfactant to avoid skin irritation, plus various ingredients that can moisturize the skin, add scent, improve lather, etc. Lathering agents are generally surfactants as well, so there's a careful balance here between getting a nice lather and not drying out skin. Hand washes are usually less foamy since they don't need to cover much surface area and are used frequently throughout the day, while body washes tend towards more foam since they need to cover a lot more surface area and are used less frequently. They're both a bit more concentrated than shampoo, since it's easier to rinse soap off of skin than hair, and in the case of body wash, most consumers pour it onto a sponge/pouf/washcloth/etc. before applying it to the skin, which spreads it out thinner than applying it directly. Face washes are their own magical category and can include all sorts of fun chemistry like ceramides and multivesicular emulsions, alpha hydroxy acids, benzoyl peroxide or salicylic acid, etc. - and as an acne sufferer, it definitely makes a difference.

Bar soaps are one area where true soaps are still relatively common. Moisturizing ingredients can also be added, and the naturally occurring glycerine is also somewhat moisturizing. However, traditional soaps also have a lot of limitations. They have a fairly narrow range of environments in which they're effective, needing hot water with a low mineral content to function, and must be rinsed a second time with clean water to avoid deposits.

Laundry detergents are commonly formulated these days with specific surfactants designed to work well with cold and hard water, both of which decrease the effectiveness of traditional soaps. They're also highly concentrated, since they're going to be diluted by the large volume of wash-water, which is why just a few drops of liquid detergent on your hands will take much longer to rinse off than an equal amount of hand soap. This is also why ideally you should fill the washer with soap and water so they can mix, then add the clothes.

Dishwasher detergents aren't worried about gentleness, since they don't come in contact with skin or organic fibers, so they can use harsher detergents and often include abrasives, but do rely on hot water to be effective. For similar reasons to laundry detergents, they're highly concentrated, but they use surfactants that are more effective on metal and ceramic.

Dish detergents meant for hand-washing dishes have to balance removing grease, starches, sugars, etc. from food with not completely stripping the natural oils from skin, which is a bit tricky - the oils that keep your skin nice and pliable aren't any different from the oils of any other animal, chemically speaking. They're also pretty highly concentrated for that "grease-fighting" effect, and so that your washcloth/sponge/etc. doesn't need more soap on it after every dish you wash.