The other week, I reported that with Google News specifically, the recommendation is that your titles and H1s be consistent.
Now a Google Webmaster Help thread has people asking if the title and the H1 (header) tags need to be exactly the same or identical.
The answer is that "consistent" doesn't mean "identical." With Google News, you probably want to keep them almost exactly the same. But with normal Google web search, having them close is fine.
Personally, I like to use a more explanatory but shorter headline with my stories here and a more keyword based title so that people who are looking for help with this topic can find it in Google. But often, the titles here and H1s here are consistent, maybe not exactly identical.
For example, the title on this post is:
Google SEO: Should Titles & H1 Tags Be Identical?
The H1 is:
Google SEO: Should Titles & H1 Tags Be Exactly The Same?
Consistent but not identical.
The main thing is for you to test and see what works for your site and your rankings and users.
luzie in the forums sums it up like this:
1. Usability - if title and headline are more or less the same, it's easier for users to identify the page they clicked through to from the result pages as the one they found on the search engine because search engine results use the title as main result element, whereas the element immediately visible on the page is the headline, the title there is invisible as long as users don't know exactly where to look for it. It may therefore irritate users if title and headline differ too much.
2. Keyword prominence - it's obvious that it makes sense to use the same keywords in titles and headlines, simply because a single page should always define it's topic as exactly as possible, which means there will always be only two or three main keywords any given page will be optimized for.
Forum discussion at Google Webmaster Help.