Got the same result (OS X) than you but this was fixed just by setting the option PlayRange->All in Play. By default this option is set to Automatic and according to the docs it says "outlying levels are dropped". This is why you observe that some amplitudes are clipped in the exported file.

Now, it is not clear why Mathematica plays it exactly the same with the option set to All or Automatic.

Hi Neil,

Thanks for replying. I have tried exporting some examples, like the one you posted, and they export fine. I can hear no pops nor can I see any clipping in the picture of the waveform. I'm only having this problem with my code. The code I'm running is somewhat computationally intensive. It takes awhile for it to render the audio file. Could this have anything to do with the export quality? Also, I have noticed that the exported file has a large DC offset (~9%), whereas the DC offset when I exported the example code was nearly zero.