What features does if offer that we can’t do today with TLS?
While the ciphersuite employed in CurveCP (i.e., Curve25519) is being added to TLS (i.e., with draft-josefsson-tls-curve25519 and X25519 in Chrome), CurveCP is more amenable to be contrasted with DTLS and other transport security protocols working with datagrams, being based on UDP.
In the vision of DJB this was designed to work with DNSCurve, where network domain names are authenticated at the packet level.