The YouTube video by Norbert Gostischa explains the differences between private browsing mode and VPNs and argues that while private browsing mode provides some level of privacy, it does not offer complete online privacy. Private browsing mode only prevents your browser from saving your history on your device, but websites and your internet service provider can still track your activity. VPNs, on the other hand, encrypt your internet connection and route it through a server in a different location, hiding your IP address and making it appear as if you are browsing from another location. This makes it harder for websites and your internet service provider to track your online activity. The video recommends using both private browsing mode and a VPN for complete online privacy, as they work together to protect your online activity from prying eyes.