Even with private browsing on, cookies are still retained for the current session and the user agent string is still sent which contains information about your device, browser and operating system.

I have noticed on occasions that BA will hold fares for a certain period if you have commenced the booking process. Therefore, if there are only a couple of seats left in a bucket and you have taken the booking beyond the search stage and maybe even selected seats if you have status, closing the bowser and starting a new search may well result in a higher fare being quoted.