The user data is stored in the profile folder. You can copy more than just the bookmarks by copying the relevant files from your backup into your current profile folder, for details see Profile backup - MozillaZine Knowledge Base or Recovering important data from an old profile.

The bookmarks file is called places.sqlite, that file also contains browsing history. You can copy that file from the backup and place it in your current profile folder to restore the bookmarks and history.

An alternative approach if you just want the bookmarks, inside the backup of your profile folder there will be a folder called bookmarkbackups. Inside that folder are files with names in the format bookmarks-yyyy-mm-dd.json, those are backups of your bookmarks. You can import one of those files into your current version of Firefox by following the restore procedure in the backing up and restoring bookmarks article, using the "Choose file" option.

Just to clarify the location, in Windows 7 it is:

c:\users\'username'\AppData\Roaming\Mozilla\Firefox\Profiles\'random letters'.default\places.sqlite

Best bet for new users is to click on Computer, then the icon of your main drive (usually C:) and then click through the folders as shown above.

Hope this helps.

I don't see the file places.sqlite under Mozilla Firefox folder

Not located in the Mozilla Firefox folder.

https://support.mozilla.com/en-US/kb/Profiles#w_how-do-i-find-my-profile