Personally, I’ve de boned within hours, and hung for after skinning for several days days. I didn’t really notice a difference in the meat.

However, my wife the taxidermist says;

If you’re going to mount it, get that hide/head off as soon as you can.
If it’s great trophy, that’s really important to you let the butcher skin if you can.

And for Gosh sakes when you field dress it.
Don’t Cut The Brisket (chest)!!!