Quick answer
Everything comes out of one and not the other. The verse is about discharge, not about whether the feelings exist.
Read both lines
The NKJV footnotes "feelings" as literally spirit, רוּחַ, so what the fool empties is his whole interior weather, and the phrase is all of it. The second line's verb is about stilling or quieting rather than suppressing, which matters: the wise man is not pretending the spirit is calm, he is settling it.
What it does not mean
This is not a rule against expressing emotion, and it is not a commendation of the person who says nothing. The book itself values speech at the right moment (25:11), commends open rebuke over concealed love (27:5), and puts a great deal of grief and anger into the mouths of righteous people across Scripture without rebuking any of it. What the verse contrasts is discharge with governance, not feeling with numbness, and a person who has been told to bottle everything up has been handed the wrong verse.
Related: Proverbs 25:28 · Proverbs 12:16 · Proverbs 27:5 · Proverbs 17:27
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.