I am a bit hesitant to write this, as I am not really familiar with parts of what I discuss,
but if I did not get something wrong I believe one can say something more or less precise, on why, as said by Henry Cohn, AKS should not be relvant for factoring. 

The AKS-primality test/prove falls into the paradigm of "derandomization," more specifically derandomization of polynomial identity testing, see for example "[On Derandomizing Tests for Certain Polynomial Identities][1]" by Agrawal, where the AKS-test is mentioned right at the start.

To put it slightly differently AKS did not use much a new or deep insight on primes or number theory, but an insight on efficient and deterministic testing of identities.

Or even more bluntly, the progress is more on the computer-science-side than on the number-theory-side.
 
While for the actual exponent in the algorithm number-theoretic results are relevant, just to get some polynomial time algorithm does essentially need no (advanced) number theory at all. [To avoid any misunderstanding, this is meant as a compliment not a criticism.]

This technique also has other applications, but factoring does not seem to be one (see e.g. the above mentioned paper).  

Finally, there are no 'fast' (for the right notion of fast) probabilistic factoring algorithms either. 


   


  [1]: http://www.cse.iitk.ac.in/users/manindra/derandomization/derandomizing-some-identities.pdf