How do quantum key distribution systems tolerate noise? It seems to me that any noise in the signal would result in Alice and Bob generating different keys, without a simple way to tell the difference. 

How do quantum key distribution systems tolerate noise?
That question is essentially A.1 of this lengthy post, with some elements of answer in B.1

For your convenience, here’s quoting those specific sections of his linked answer:


  A. Issues prevent direct use of sifted bits
  
  …
  
  
  Errors creep in the sifted bits
  
  The many imperfections of the model (heavily simplified compared to the actual hardware and the known laws of quantum physics, even at the time the [article][2] was written) are such that, without Eve messing, sizably many sifted bits differ at Alice's and Bob's end. Sifted bits are thus unusable as key to a [cipher][3], for decryption would likely fail.
  
  
  …
  
  B. Classical cryptography rooted in information theory fixes them all
  
  
  Secret-Key Reconciliation protocol remove errors, and more
  
  Coding theory and error detection and correction are used to build a secure Secret-Key Reconciliation protocol additional to sifting, also running over the classical channels. Its objectives are to
  
  
  remove or fix errors in the sifted bits, with bounded low odds of the contrary;
  obtain an upper bound on the actual error rate, which is attributed to Eve spying the quantum channel;
  obtain and minimize an upper bound on the information leaked by running the protocol over the classical channels, that Eve is assumed to scrutinize;
  deduce a lower bound of what entropy (if any) remains in the rest; if there is less than some headroom, QKD failed.
  
  
  Such reconciliation protocol can loose little-enough entropy as to be usable in practice, and be information-theoretically provable; but no simple such one is known. A most scrutinized milestone is the cascade protocol in: Gilles Brassard and Louis Salvail's Secret-Key Reconciliation by Public Discussion, in proceedings of Eurocrypt 1993.
  
  
  …