Also, Firefox container tabs might be a nice solution: normally browse in a container where I'm logged out, then to edit, change the tab to my Wikipedia container where I'm logged in.
The main things that are different:
* the user links (shows what user you are logged in)
* whether you have a new message notification
* your skin preferences (this will totally change the entire html)
* your language preference (normally this only affects the UI not page content, but some pages it affects page content)
* other misc preferences can affect wikipage content (e.g thumbnail size), although there has been a general effort to avoid adding new ones and remove existing ones that arent used.
* some stuff makes assumptions about logged in users not being varnish (e.g. how csrf tokens work for js based actions), but most of those can be changed if push came to shove.
That said there might be a reasonable argument that much of this isnt needed and it would be a net benefit to kill most of those features.