For example, bar soap uses a different soap chemical than liquid soap (typically sodium stearate vs. typically sodium laureth sulfate). It’s a lot better at removing oils/grease, therefore better at cleaning but will also leave your skin dry.
And then there’s shampoos, which will have additives for your hair. For example, various shampoos for longer hair are prepared in such a way that when they get mixed with water, silicone will fall out of the shampoo and stick to your hair, giving it a shine and making it stick together less.
Well, and for completeness’ sake, there’s also cleaning detergents which technically contain soap, but you really don’t want near your skin.
But yeah, having said all that, any soap intended for use on your body, should be safe for skin and hair. If you intend to switch long-term, then you should watch out for dry skin or less healthy/beautiful hair.