What is the difference between killing Poro with Hidden Zhonya in the Showdown and outside Showdown?

The difference is when your last chance to flip Zhonya's happens — and whether it can actually save the Poro.

In the Combat Showdown: once both players pass, the Showdown Closes and damage is assigned and dealt with no window for reactions (Rule 465.2). The Poro dies during Combat Cleanup. By the time its Deathknell opens a reaction window, the death has already happened — and Zhonya's is a replacement effect that must be active before the death to replace it (Rule 369.1). So a still-hidden Zhonya's cannot save the Poro from combat damage. Your only shot is flipping it during the Showdown Open State, before you pass — which reveals it and lets your opponent assign damage accordingly. (Flipping it in the Deathknell window still saves the Zhonya's itself from being trashed if you lose control of the battlefield.)

Outside the showdown: if the Poro is killed by a spell or ability on the chain, that creates a Closed State — you can flip the hidden Zhonya's as a Reaction before the kill resolves. When the kill lands, Zhonya's is face-up and active, so it replaces the death: Zhonya's is killed instead, the Poro is healed, exhausted, and recalled to base, and no Deathknell triggers. The one exception is a death that happens during a cleanup with no prior chain window (e.g., lethal damage already marked) — there you're too late either way.