Do Google Adwords show up with a slightly different referrer than the organic traffic? I know you can track with tracking codes using various software apps from ClickTracks all the way up to Omniture, but can you tell just by looking at the naked logs?
The only way I've seen is when google sends you an AdSense referral - they all appear to come from the domain googlesyndication. Thats why tracking query fields or a special google entry page are the best ways to do that.
If you think about it, having the same referrer does make sense as either way they are clicking on a google web page, either in search results or from an ad.
Can you explain "organic traffic", in your original question?
If by "organic" you mean unpaid clicks (vs paid clicks), there are only 2 ways to know and both involve setting up the ads so they can be tracked. 1)is if you set up the original ads to have a tracking parameter in the URL, the way Overture will do for you automatically if you turn URL tracking on. 2) is if all your AdWords ads point to some kind of landing page that is only used by AdWords ads.
If by "organic" you mean Content Match vs search engine match, the "googlesyndication" method can do it, I think, but it's just as easy to examine the list of referrers directly and separate the search engines from the others.