IPv6 is the next generation of the Internet Protocol that is being deployed around the world to replace IPv4. In the design of IPv6, extension headers allow the protocol to be flexible, enabling optional features, such as fragmentation or encryption. However, the complexity of this design often leads to vulnerabilities that can affect millions of hosts worldwide. In this paper, we propose a new methodology that exploits differential fuzzing to uncover and analyze vulnerabilities in IPv6 network stack implementations. We run a thorough set of experiments to validate the methodology on several operating systems, such as Windows, Linux, and FreeBSD, uncovering two firewall policies bypass and multiple fingerprinting possibilities. The results highlight the danger of vulnerabilities at this level of the network stack and underscore the importance of testing IPv6 as it becomes the core part of the Internet. Researchers and implementers can use our proposed methodology to look for security issues in other operating systems in a semi-automatic way.