Cryptographic network protocols play a crucial role in enabling secure data exchange over insecure media in modern network environments. However, even minor vulnerabilities can make protocols an easy target for cyber attackers. Therefore, it is essential to investigate the threats and vulnerabilities stemming from the cryptographic network protocols. Furthermore, it is necessary to comprehensively investigate the weaknesses of network protocols that use cryptographic primitives to inform users and developers about potential attack points. This comprehensive survey examines the relationship between encryption schemes and network protocols and presents an in-depth review of associated threats and vulnerabilities. Given that most cryptographic protocols operate in the Transport and Application layers of the Transmission Control Protocol/Internet Protocol (TCP/IP) protocol stack, our investigation primarily centers around encryption algorithms used by representative and notable cryptographic network protocols such as Transport Layer Security (TLS) and Secure Shell (SSH). Furthermore, we delve into the attackers' methods to exploit the already identified and existing vulnerabilities, seeking to understand the mechanisms employed to compromise these protocols. Through this survey, we aim to provide the readership with an in-depth understanding of the existing and new vulnerabilities associated with modern cryptographic protocols and provide valuable insights into securing them effectively. We also discuss the existing challenges and future research directions in this domain.