Audio Sync

In the help section for Audition it mentions the program will snap audio lines if you used SMPTE time code. I bet you didn't. I suppose you can use Auditon multitrack editing, but it won't sync automatically you'll have to line the peaks up visually. Look for "Importing video and working with video clips."

Audition is an audio editor, not a video editor. Pretty much any NLE these days will allow you to sync an external audio file to your existing video/audio file. Hopefully you used a slate or have a definite spot that you can use to sync the external audio.

Mike

If you edit in Premiere, then you are normally limited to shifting the audio by a frame as the smallest amount. If your bring video into the audio editor, you get better audio accuracy, but of course few video features. Personally, I find the accuracy of slipping the audio in Premiere sufficient for most projects. 

I have never had the grief some people report syncing audio to video – it's just a case of finding suitable hit points – door slams, noises, maybe a click of the lightswitch, or in music projects, the drummer, or perhaps keyboard player. For those that find this difficult, Plural eyes, in your video editor  does it for you by comparing the audio waveform with the camera audio waveform. Some people seem to need it, others have the right ear and can do it themselves.