In the quantum teleportation protocol Alice can send Bob an unknown quantum state $|\psi\rangle$. If the only thing Bob does with $|\psi\rangle$ is to measure it in some basis, I guess it would be simpler for Alice to just do the measurement herself and send the result to Bob. That would not require a shared entangled pair but Alice would need to know Bob's favorite basis. Is there something Bob can do with an actual state $|\psi\rangle$ which can not be done with information that Alice sends to Bob?
More generally, due to Holevo bound only a single bit of information can be encoded in a qubit. What is then the difference between teleporting an actual quantum state $|\psi\rangle$ and sending the result of some measurement of this state?