Learn English – The difference between “retail” and “sell”.

Semantically they're very similar, but syntactically, the difference is that underlying is an adjective, whereas underneath is either a preposition or an adverb.

The syntactic difference means that, like most adjectives, underlying usually occurs before the noun it modifies...

This sentence illustrates my underlying point.

...but note that we can also use it as a "continuous verb form", with "point" as the subject...

The point underlying that sentence is made again here.

Apart from that syntactic difference, note that the relatively less common underlying usually refers to something fundamental, that underpins something else, whereas underneath (and plain under) merely implies something being (physical or figuratively) below something else.

Cold-hearted carries more of a connotation of being unfeeling and lacking in empathy - the ability to share, understand and feel another person's feelings - whereas hard-hearted carries more of a connotation of lacking in sympathy - an understanding of and feeling for the sadness or suffering of others.

Cold-hearted is more of a passive condition, of being uninterested in the feelings of others. Hard-hearted can be more of an active condition, of being disinterested in the suffering of others.

And there is a very large overlap of synonimity between them.