Practical consequences of a honest-verifier zero-knowledge of the Schnorr protocol
By definition, a dishonest verifier may be able to completely extract the witness from an honest-verifier ZK proof.

To prove this: take any zero-knowledge proof, and add an initial round where the verifier sends 1 to the prover. The prover then works as follows: if it receives 1 then it runs the original zero-knowledge proof; if it receives 0 then it sends the witness.

The above proof is clearly honest-verifier zero-knowledge, but a dishonest verifier can learn the witness by sending 0.

Typically, the point of an honest-verifier zero-knowledge proof is that it can be "compiled" into a (full) zero-knowledge proof without much difficulty. 

Having said the above, some honest-verifier zero-knowledge proofs already have stronger properties like witness indistinguishability. This makes no sense for Schnorr itself since there is only one witness, but I believe that the standard Diffie-Hellman tuple sigma protocol is witness indistinguishable (I would want to write a proof to be 100% sure but I'm nearly sure).