You're slightly mistaken about the mechanism for teleportation. The process does involve sharing an entangled pair of particles, but the individual states of those particles are never measured. Instead, a joint measurement is made of the state of particle whose state A wants to teleport to B and one of the two entangled particles-- basically, asking "are you in the same state, or different states?" Based on the outcome of that measurement, which is sent to B via a classical channel, B does one of a small number of simple operations to the other entangled particle, which leaves it in exactly the state of the original particle that was being teleported.
The advantage of this process is that the state of A's original particle is never directly measured, which means that it arrives at B completely intact. This is the only sure way to transmit a single unknown quantum state without sending the particle in that state directly. You might think you could just make a copy of the state, and send the copy, but that is forbidden by the "no-cloning" theorem-- it is impossible to make an exact duplicate of a single quantum state unless you know in advance what that state is.
The key to the process is that the entangled pair of particles do not have well-defined states, but are correlated in a non-local fashion. You're not sending a set of random numbers with definite values from one place to another, you're sending an indeterminate state whose exact value will not be determined until it is measured, at which point its value will absolutely determine the state of a second particle a long distance away.
The process is of interest to people studying quantum information because it may be the surest way to move quantum information from one place to another. It's not going to allow you to build a Star Trek transporter any time soon, but it might be useful for building a quantum computer, or for connecting together quantum computers at distant locations.
TL;DR
Teleportation plays an important role in the theoretical analysis of quantum cryptography. Most of the security proofs work by reducing the considered quantum cryptography protocol, which may or may not be entanglement based, to an equivalent entanglement based protocol, which uses quantum teleportation.
Theoretician's teleportation based QKD protocol
Let's start with a "theoretician's quantum key distribution (QKD) protocol", which is involved, impractical, but the security of which is easy to prove. This protocol would would work as follow :
- Alice and Bob share many pairs of entangled particles
- They select randomly a subset of them and make local measurements to check that their particles are in a state close to maximally entangled state. (if not, they abort)
- Alice then encodes a random string in one set of particles, for example in the rectilinear basis of the polarization of photons
- She teleports the particles to Bob, which measures the particles and get the secret information (the random string)
Because of step 2. Alice and Bob knows that the entangled state is close to a maximally entangled state, and because of the monogamy of entanglement, that this state cannot be correlated with anything else, in particular with anything under Eve's control. The global quantum state is said to be factorized (i.e. under the form $\left|\Phi^+\right>_{AB}\otimes\left|\psi\right>_E$), and Eve is "factorized out". This allows to easily show that the step 4 does not leak any information to Eve, and that the protocol is secure.
You can also consider using quantum error correction codes in step 1 to add some quantum computing in the mix. It's a theoretician's protocol, after all !
Reducing entanglement based QKD protocols to the previous one
Some QKD protocols are entanglement based. The most famous one is Ekert's E91. They work as follow :
- Alice and Bob share many pairs of entangled particles
- They make random measurement on them
- They reveal a subset of measurement that the state is close to an maximally entangled state and abort otherwise.
- Since the state is close to a maximally entangled state, the other measurements are perfectly correlated between Alice and Bob, and not correlated with anything else. Eve has been factored out.
The previous protocol works well, doesn't use teleportation, but has a big problem : as soon as the channel used in step 1 and/or the detectors used in step 2 are slightly imperfect, the state is slightly different from the maximally entangled state and Alice and Bob aborts. That prevents any use of it in any realistic scenario.
The solution is to use some classical error correction (information reconciliation in QKD parlance) and privacy amplification (usually based on universal hashing). These information theory based techniques allow us to distribute secret keys if the imperfections are not too big, but they rely on relatively complex data processing which is not easy to analyse theoretically. However, it can be shown that the whole protocol, including this classical post-processing, is equivalent to a teleportation based protocol where the step 1. was made using the relevant quantum error correcting codes. This equivalence allows to prove the security of a practical QKD protocol by reducing to a teleportation based protocols, which is impractical but theoretically easier to analyse.
What about entanglement-less QKD protocols ?
Most practical QKD protocols, and among them the 1st protocol, BB84, do not use any entanglement. They are prepare and measure protocols (P&M) where Alice choses randomly to send the state $\left|\psi_i\right>$ with probability $p_i$, and Bob performs a measurement. If Alice choice is perfectly random (and we assume it is), such a protocol is perfectly equivalent to a protocol where Alice prepares the entangled state $\sum_i\sqrt{p_i}\left|i\right>_A\left|\psi_i\right>_{A'}$, sends the $A'$ particle to Bob and measures the $A$ particle in order to get $i$. So we're back to the previous paragraph and the teleportation based protocol.
Why reducing this the simple protocol (the last one) to a more complex one ?
The main reason we go through these equivalences, is that the teleportation protocol is easier to analyse. More specifically, we can show that no one but Alice and Bob can get any information in the process without having to know much about Eve's attack. In particular, we do not have to enumerate the set of attacks Eve could perform and take the risk to forget one attack.
In short, the teleportation based protocols are easy for theoreticians but a nightmare for experimentalist (you need quantum computers to perform them with error correction !), while the P&M protocol are easy (or at least doable) for experimentalist, but difficult to analyse directly for theoreticians. Except, of course, when they are shown to be equivalent to a teleportation based protocol !
Best Answer
The main difference between quantum cryptography and quantum teleportation is the following :
So the analogy between quantum teleportation and quantum cryptography is very deep, and not a mere coincidence.
$*$ I know that some quantum cryptography protocols have other aims than privacy. But I don't think it's the subject of this question.