Vault vs. Chef, Puppet, etc.
A big part of configuring software is setting up secrets: configuring a web application to talk to a service, configuring the credentials of a database, etc. Because of this, configuration management systems all face a problem of safely storing these secrets.
Chef, Puppet, etc. all solve this in a similar way: single-key encrypted storage. Chef has encrypted data bags, Puppet has encrypted Hiera, and so on. The encrypted data is always one secret (a password, a key, etc.) away from being decrypted, and this secret is generally not well protected since in an elastic environment, every server needs to somehow get this secret to decrypt the data. Additionally, access to the encrypted data isn't always logged, so if there is an intrusion, it isn't clear what data has been accessed and by who.
Vault is not tied to any specific configuration management system. You can read secrets from configuration management, but you can also use the API directly to read secrets from applications. This means that configuration management requires fewer secrets, and in many cases doesn't ever have to persist them to disk.
Vault encrypts the data onto physical storage and requires multiple keys to even read it. If an attacker were to gain access to the physical encrypted storage, it couldn't be read without multiple keys which are generally distributed to multiple individuals. This is known as unsealing, and happens once whenever Vault starts.
For an unsealed Vault, every interaction is logged in via the audit devices. Even erroneous requests (invalid access tokens, for example) are logged. To access any data, an access token is required. This token is usually associated with an identity coming from a system such as GitHub, LDAP, etc. This identity is also written to the audit log.
Access tokens can be given fine-grained control over what secrets can be accessed. It is rare to have a single key that can access all secrets. This makes it easier to have fine-grained access for consumers of Vault.
For tips on how to integrate Vault using configuration management, please see Using HashiCorp's Vault with Chef. Although this post is about Chef, the principles can be broadly applied to many of the tools listed here.