How to close the emoji palette using the keyboard shortcut?
The behaviour is somewhat awkward.

Even the Menu option doesn't toggle it on/off, only on.

If you simply double-click your emoji without doing anything else, it doesn't even bring the window to the front... 

..so, the only real option is after you open it, click its header bar [while you're over there anyway to grab your emoji]
It will then respond to  Cmd ⌘   W  the same as any other window.  

 Of course, if you forget, it will close the window you were working in  :\