This is one of those things I always thought I understood. The language is easy enough, but when I got down to it, I couldn’t explain what save ‘as a copy’ meant in Photoshop. Doesn’t ‘Save As’ always save as a copy of the original document? Adobe puts it this way,
As A Copy Saves a copy of the file while keeping the current file open on your desktop.
but that seemed vague before I knew what it meant. Here is a more personable explanation in my own words:
‘Save As’ without checking ‘As A Copy’ saves the current file in the format you select and then makes that resulting file the current file open in Photoshop (what Adobe calls ‘your desktop’). ‘Save As’ with checking ‘As A Copy’ saves the current file in the format you select but leaves the original file as the current file open in Photoshop. In other words, choosing ‘As A Copy’ is like saying,”I’ve been editing you for a while now, and I want to save my progress in a separate file, but I’m not done with you, I want to continue editing you after I save.”
This would be important to choose if you want to ‘Save As’ a JPG or some other compressed format because you don’t want the compressed file to be the one you continue editing. Anyway, that’s my opinion. Take what you will.