Ben Morei stated the difference between HTML4 and HTML5 in using heading tags already. His answer is therefor completely correct.
What i would like to add to his answer is that you should look at H1, H2, H3, H4, H5 and H6 tags as if the webpage is a book. In a book the chapter-title is a H1 tag. All the subchapters have an H2 tag. If these subchapters have another sub-area then this would be marked up with an H3 tag and so on.
Having a clear hierarchy in your heading tags will make it more user friendly. Search engines might also understand better which parts of the content are connected to each other and which are separate because of this.
Hope i answered your question and if anything is still unclear feel free to ask more details.
We have three answers to your question here now and the question is still on marked as 'unanswered'.
If your question is answered could you please mark the answer which was most helpful as the correct answer so this question is no longer on the bounty page for no reason?