In identity-based encryption (IBE) the public key of a user $A$ is some identity string $ID_A$, e.g., an email address. To encrypt a message for this user there is no need to retrieve an authentic copy of the public key, as it is in public key cryptography, but one can simply use the identity string $ID_A$.
Identity-based cryptography (IBC) aims at letting the users public key a public identity string, e.g., the email address, and so to remove the requirement for certificates. In ID based cryptography a user $A$ uses his identity (e.g., his email address) as public key. A trusted authority (the key generation center) computes the private key $sk_A$ that corresponds to the public identity string $ID_A$. Since the private key is generated by another party on behalf of the user, IBC suffers from an inherent key escrow problem. However, there are possibilities to circumvent these issues.