And “difficult” is used when we define “hard” as in “difficult to do, understand, solve or explain”.

I see them on a spectrum like

simple – easy – difficult – hard – draining – impossible

For me “difficult” is easier than “hard”. A “difficult” test is not as challenging as a “hard” test.

There is also a connotation that “hard” is a little more physically demanding / stressful “it was hard to climb that mountain” (meaning that your muscles were very tired by the time you got to the top) where as difficult is, as you say more mentally challenging, if it was difficult to climb the mountain, it was tough to pick the right route, or know where to set the ropes.

None of this from any book, just my own impressions as a native speaker.

examples..

That food is too hard to eat..

The ball hit me hard..

This bed is too hard to be comfortable.

This quiz is too difficult..

This is a difficult operation..

This is a difficult dish to cook..

the word hard can mean difficult, but difficult can never mean hard.. what I mean by this, is that you can change the word difficult to hard in any sentence but not the other way around in every sentence.