Is there any part of OSX that gets a significant speed boost from Intel AES instructions?
So... A bit of poking lead me to a simple conclusion: it's unlikely Apple will tell you where they use AES-NI instructions within Mac OS or the rest of their software, but they are definitely present and available.

Firstly, Intel talks a bit about AES-NI and where it is used. Some of the comments talk about its presence or lack there-of in the comments. Given that I have homebrew and wine installed on my Mac, I followed the interesting instructions here to test if the openssl dependency of wine is able to use AES:

$ openssl speed aes-256-c
...
The 'numbers' are in 1000s of bytes per second processed.
type             16 bytes     64 bytes    256 bytes   1024 bytes   8192 bytes
aes-256 cbc      69308.49k    74198.23k    75165.95k    74875.63k    76098.51k

$ openssl speed -evp aes-256-cbc
...
The 'numbers' are in 1000s of bytes per second processed.
type             16 bytes     64 bytes    256 bytes   1024 bytes   8192 bytes
aes-256-cbc     421057.71k   432531.16k   447445.08k   450021.03k   444866.56k


These numbers broadly agree with cyberciti's quote about 6x performance improvement using AES-NI. 

Finally, we have some actual evidence in the SE community itself: https://stackoverflow.com/questions/12634305/.

Apparently, Apple used to provide an API for Mac OS which would not allow direct use of AES-NI instructions. The work-around used to be using Intel's compiler or building gcc from source. It appears that since 4.6.3, this is no longer the case, and developers can make use of AES-NI if they so choose. The performance improvements are then a function of the developers, not the OS.