The difference between soaps and syndets comes down to how each is made, and that manufacturing difference is what drives every practical advantage syndets offer.
Soap is produced by a chemical reaction called saponification, where a fat or oil is combined with an alkali such as sodium hydroxide. This reaction always leaves the finished soap with a naturally alkaline pH, typically around 9 to 10, because the alkali used to make it cannot be fully removed from the final product.
Syndets, or synthetic detergents, are built differently. Instead of coming from a fat-plus-alkali reaction, they are manufactured from petroleum or plant-derived synthetic surfactants that chemists can engineer to almost any pH they choose, which is why syndets are typically formulated close to skin's natural mildly acidic pH rather than being stuck at an alkaline one.
This same manufacturing origin also explains their behaviour in hard water. Soap molecules are fatty acid salts that react with the calcium and magnesium ions present in hard water, forming an insoluble scum that reduces cleaning power and leaves residue. Syndet molecules are built to resist reacting with these minerals, so they keep working effectively regardless of water hardness.
Therefore, syndets give better results than soap because the way they are manufactured, as engineered synthetic surfactants rather than a fat-alkali reaction product, allows their pH and mineral resistance to be controlled, while soap is chemically locked into being alkaline and reactive with hard water.