Quick answer
One man's anger is immediately public. The other absorbs an insult without broadcasting it, and the verse counts that as prudence.
Read both lines
The contrast is about timing rather than about feeling. Nothing here says the prudent man is not annoyed; it says his response is not instantaneous, and the gap is where judgment happens. "Covers shame" is the interesting phrase, because the shame in question may be his own or the shame of whoever caused it, and both readings are useful. Choosing not to make an incident into an event is a real skill, and the book treats it as one, the same way it treats the man of understanding who holds his peace at 11:12.
Related: Proverbs 11:12 · Proverbs 29:11 · Proverbs 19:11
New King James Version®, Copyright © 1982, Thomas Nelson. All rights reserved. Full copyright information. Scripture provided by API.Bible.
Go Deeper
If this verse has been used to put weight on you that it was never carrying, ask about it directly.