Hashicorp Vault is a strong, enterprise-focused solution for secrets management. But despite its focus on security, its lack of end-to-end encryption is a significant drawback in comparison to EnvKey. Ultimately, regardless of how secure your Vault server setup is, it still has access to all your secrets in plaintext, giving it a much larger attack surface than EnvKey. It doesn't do much to address insider threats, paricularly those that originate on a DevOps team. Any engineer with operational access to the Vault server will necessarily have to be trusted with access to all an organization's secrets.
Vault's complexity also introduces an ongoing maintenance burden and 'integration tax' that can slow down development and deployment processes. EnvKey's focus on developer productivity and ease of use makes it much easier to setup in the first place (minutes vs. weeks), and makes loading secrets for any environment, from development to production, something you'll rarely have to think about.
While Vault has some features that EnvKey lacks, like encryption-as-a-service and LDAP integration, overall EnvKey offers a simpler and more secure approach to configuration and secrets management for most organizations. And though Vault is well-established and trusted in the enterprise, EnvKey is no slouch on enteprise-focused features, offering high availability, audit logs, SSO, SCIM, teams, role-based access control, and scalability that can handle thousands of developers and servers.
Vault has (deservedly) reached "nobody gets fired for choosing it" status in the enterprise, and it could certainly make sense to reach for it if you need some of its more advanced features, but for many organizations, EnvKey will win out in a cost-benefit analysis.