The patch I’m working on now will change the current behavior.
Currently, all requests from running the client are treated as independent and so it will obtain a completely new cert and save it in a different place (you’ll have something along the lines of domainname.com-1 and domainname.com-2 directories, one for each time you ran the client, with separate private keys and separate certs). The process will also create a new private key, and it will update the web server config to use the new cert, wherever it’s applicable, in place of the old one.
My patch will do a “superset detection” and prompt you to replace the old cert with the new one. However, that will just be done by updating the symlinks; the old cert won’t be deleted, but anything that pointed at the symlink for the old cert will now point at the symlink for the new cert. And probably we will use the same private key. In any case, the web server configuration files won’t be changed. This is also called “treat as renewal” and responds to LE client issue #411.