If you need Firefox to remember new or revised passwords you must set it to remember History rather than be in private browsing mode. See Usernames and passwords are not saved. You must also avoid the custom setting to clear History When Firefox closes or again new passwords will be lost.

You are able to manually clear some of the specific History items without loosing passwords.

• Use Firefox Button -> Options -> Options |Privacy| History - clear recent history
• see Settings for privacy, browsing history and do-not-track

If you select "Never Remember History" then you enter Private Browsing - Use Firefox without saving history mode and in PB mode Firefox doesn't offer to store new passwords. Also all cookies are session cookies that expire at the end of the session or if you close Firefox, so websites do not remember you.