In my experience: there is no significant difference as far as the search engines are concerned. I agree with Rand in that for many users, the first thing they will expect to see is the same title they clicked on via the search engine.

My rule of thumb is always do what makes sense to users. Even if a different H1 tag caused some significant affect such as moving you up on the SERP (unlikely), if the user doesn't get what they expect they will click back. If anyone experiments with this, pay close attention not only to ranking but also to bounce rates. I'd rather be the second result with a 10% bounce rate than the first result with a 90% bounce rate.