The difference is that traditional Connect is typically a two party model - an OpenID Provider which gives claims which can be used for registration/authentication, and a Relying Party willing to accept them. This is an active dance back and forth, with the OpenID Provider deciding how to implement privacy, what records to keep on usage, prompting for user consent, etc.
Verifiable Credentials have Issuers and Verifiers which map reasonably well into these roles, but also an end user agent in the middle which acquires credentials, holds onto them and presents them with user consent.
While the verifier needs to still know who the issuer is to know if it should trust the data, the issuer no longer needs any relationship whatsoever with the verifier. The issuer does not see where credentials are being used, or (for credentials with selective disclosure) which information is being disclosed.