Private browsing creates a temporary and separate profile. During private browsing, all private tabs and private windows can share cookies (not with standard browsing tabs and windows).

If you are login to Facebook in standard browsing, theoretically Facebook can't know which sites you are visiting in private mode. (or opposite: login to Facebook in private mode and visiting sites in standard mode)

But who knows if Facebook is not using IP addresses to track users? Actually they do, you can check it on this page: for "security reasons", Facebook saves data about your previous connections. And if Facebook knows your location, that means it uses IP addresses.

Private Browsing is related to what is not saved in Firefox after the Private session is closed. "Private" as to how it relates to your own PC recording where you go and what you do on the internet, and getting rid of that information before someone else uses Firefox on that PC after you're done using it. https://support.mozilla.org/en-US/kb/private-browsing-use-firefox-without-history

Not tracking users has to do with the Do Not Track features in Firefox as used by websites who may [or may not] honor DNT standards in web browsers. https://support.mozilla.org/en-US/kb/settings-privacy-browsing-history-do-not-track

If you log on to a website in PB mode using the same user name (account) as in normal mode then that doesn't make a difference for the website server because they identify you via the user name that you use to log in. Only data like cookies stored locally in the profile folder is not accessible in PB mode.

During a PB mode session all cookies are kept in memory. In PB mode a separate cookie jar is used and cookies saved in normal mode can't be accessed.

If you see the Facebook Like button on a page, Facebook knows your copy of Firefox visited the page. The cookie associated with the request for the button lets Facebook know whether it has seen this particular Firefox before. Whether Facebook knows it is you depends on whether Facebook can associate that cookie with your account, which normally would only happen if you logged into Facebook using that cookie. In a private window, you have a fresh set of cookies, so if you avoid logging into Facebook, you may break the link.

It's also possible that if Flash is involved, there may be a privacy-bypassing Flash Local Storage Object (LSO, Flash cookie, or super-cookie) which can identify you even if regular cookies are not available. It's all rather sneaky.