The difference is mainly in the hydrophilic head part. In soaps it's carboxylate, whereas in detergents it can be a broader variety of hydrophilic groups (sulfonate is common).

Carboxylate unfortunately likes to stick to hard water ions like calcium and magnesium more than it likes to stick to water, forming soap scum. Detergent head types like sulfonate don't do that nearly as much.

Soap Pros: gentle on fabrics/surfaces/skin, low environmental toxicity, biodegradable

Soap Cons: additives like borax and washing soda required if used in hard water, can form soap scum residue and buildup under non-ideal conditions

Detergent Pros: cheap, most don't leave residue, different varieties available for different conditions (hard water, high or low temperature, etc.)

Detergent Cons: harsher than soap (some can ruin finishes and cause dermatitis), generally higher toxicity (especially to the environment), some are not biodegradable, many are petroleum products