If you're running AdWords campaigns for your Business anywhere around the globe, you need to be using this tool for the following reasons:
1) Google Recommends That You Use the AdWords Ad Preview Tool!
Google actually recommends that you use the AdWords Ad Preview Tool to check your ad positioning for keywords you are bidding on.
If you use the normal Google.com page and search for your targeted keywords, you run the risk of artificially increasing your impressions. Since you aren't likely to click on your ads while conducting the search, you're also going to be lowering your clickthrough rate with each search. This can hurt your quality score and your ad positioning. It may cost you more to maintain your current ad positions. So, you really need to be using the Google AdWords Ad Preview Tool instead of Google.com to check your ad performance for certain keywords.
2) You Can Check Your Ads in a Different Geographic Region
If you're in Miami and you're running ads in Los Angeles or New Yourk, there is no easy way to see your ad as a person in Los Angeles or New Yourk would see it. Sure, you'll get to see the average position from the campaign stats, but the AdWords Ad Preview Tool lets you act as if you were a resident of Los Angeles or New Yourk. You'll see the actual sponsored listings. Perhaps even more valuable to your business - you'll also get a good look at who is competing with you for the same keywords.
3) Advertiser Competition Analysis
If you're thinking about entering a specific geographic market with an ad campaign, the AdWords Ad Preview tool will let you know how many advertisers are bidding on your coveted keywords. You can see the total quantity of advertisers, and click on advertiser links to get a better idea of the competition that you'll be up against if you decide to enter this new geographic market.