Barry Schwartz has already posted a good overview on what the search-based keyword tool does last week. In a nutshell, the new tool shows you what the most popular keywords related to your site are that you are not bidding on, and suggested landing pages to go along with each keyword suggestion if you have access to the Google Adwords account for a given site.
The Ad/Search Share feature you see when logged in shows you what percentage of time you appear in paid search and organic listings which every PPC and SEO marketer loves to know.
Without Adwords or Analytics account access, you will only see a sampling of 100 keywords related to the site's pages. As Barry mentions, that might leak what terms a website is bidding on to competitors.
How to Apply Google Search-Based Keyword Research
Pay Per Click Campaigns
Obviously, you can use this tool to identify new keywords to bid on in pay per click but I caution you, Google's suggested landing pages are not always the best ones:
For example, "Bob Marley t-shirt" gets 6,600 searches per month so Google suggests a landing page for BustedTees.com which is actually for a Bob Marley poster, not a t-shirt.
Identify SEO Problems
If BustedTees did carry Bob Marley t-shirts, and Google suggested the poster's page as the best match, that could be an indication that Google can't "see" the Bob Marley t-shirt page, indicating a crawling and indexing problem, which may be affecting more pages on the site.
New Product Ideas
BustedTees may consider adding Bob Marley t-shirts to its catalog.
Another great way to use this tool is to prioritize which landing pages to optimize if you want to focus on terms with the highest potential keyword volume, not necessarily the most keyword referrals you actually receive (which your analytics would tell you). You could optimize for better search engine rankings (and increase your traffic) and conversion (to maximize ROI) together. A keyword research tool alone doesn't match keywords to your specific site, so this is helpful that Google does both.
You should also give your internal site search tool a quality check using the most searched for keywords. How people search in Google is often what they type in your site search box. Do you deliver the right, relevant products customers want to see or does your "searchandizing" need optimizing?
Use Your Noodle
As with any SEO or keyword research tool, use your judgment whether the suggested keywords (and landing pages) are relevant to your business. So what if "cheap cologne" gets 56,000 searches per month if you sell luxury brands? Or what if the average hides a seasonal spike in February? Do you want to pay more to advertise for cheap cologne in September?
So, my verdict is this tool is great if you're logged in to a Google Account for your site and can have application beyond PPC, but is not so helpful for general keyword research or competitor research.