I can attest to that. I originally had a page with bold text, which I later changed to H1 (as part of optimization effort
) The page actually dropped in ranking... until I changed it back to the bold.
Now, this is a pure speculation, but here is what I thought:
Header tags are originally meant to define hierarchy on the page (e.g. H1 being the title, h2 defining categories, h3 defining sub-categories etc.). H1 on its own doesn't really define hierarchy and kind of looses its purpose, in which case I can see why Google would be putting more importance to bold text (since it is bold text that's used for making certain parts of text stand out). I hope everyone follows my thought...
If you want to extend your experiment to test this hypothesis, try adding some structure to your first page with H1, H2, and H3 tags, while repeating the same layout on the second page using bold text instead for headings.
...and please don't forget to share your findings!