what's the difference in meaning between an adjective and the structure "noun + of + article + noun"?

Example:

Did you hire that clown of a teacher?

and

Did you hire that clownish teacher?

Or

My idiotic friend

and

My idiot of a friend?

Solution 1:

The two variants of each example you give are more or less synonyms, but the structures with of a are more informal and convey more about the person who uses it.

Of a is an idiom which can also be combined with other parts of speech, not only with nouns:

US, informal —used to indicate that someone or something is a particular type of person or thing:

• her idiot of a husband [noun + of a + noun]
• It is not that much of a problem. [adverb + of a + noun]
• How big of a piece do you want? [adjective + of a + noun] (M-W)

As for clownish teacher and my idiotic friend, they are less informal and more neutral. The structures with of a also point to feelings in the speaker such as anger, irony, irritation. It can also express positive feelings like admiration:

• an angel of a girl [an angelic girl]
• a beauty of a woman [a beautiful woman]
• a mountain of a man [a man as tall as a mountain]