If your main content is about the same, just make the switch. Don't give your users two versions of youyour website. Those are your main concern, Google will follow the users.
Moving your old site to a new directory makes even less sense: Youyou have a PR per page, or actually per urlURL. If you change /some-page
to /backup/some-page
, you will looslose that the PR and have the backup indexed again. Which you don't want, you want your new site to take over.
Short answer: Just make the switch. Just make sure all your urlsURLs are preserved, and try to use as many 301-header as you can where needed.