That was a hard one 🙂
The problem comes from the header image alignment.
On the broken site, it appears the Header image is set to align left, causing the image to be floated left, and that makes the following widget area expand up and cover it. I am not quite sure why the header widget area expands up over the header image in that configuration, but if you change the header image alignment to center or no alignment, it should fix it.
Can you check on the old site (good one) what the header image alignment is set at and confirm.