In editing mode, you have the website builder tool bar on the left. So the website does not actually spend full width. When you publish the site, the tool bar will be gone so the site is display wider than in editing mode. Since the header spend the full width of the page, it will look bigger once the left hand side tool no longer there. This can make the image look a little different compared to the publish site.
In addition, Weebly’s image perfect editor automatically crop your header image according to the size of the monitor. This can further change the shape of your header image. One way to help adjust the header image area cropping area is by adjusting the width of your browser: