Hello Joachim,
Since your browser-facing folder for WeeWX is /public_html, you should specify the logo using a server-relative URL:
[Extras]
logo_image = "/public_html/logo.png"
This works because the browser will fill-in the scheme, hostname and port so there's no need to fully-qualify the URL.
On a separate note, the public_html folder is an internal path and not normally exposed directly to browsers. One common method is to have the pages exposed as the folder /weewx which may be aliased to the folder .../public_html/weewx. If you have any questions on this topic, please start a new thread and mention if you're using Apache or Nginx and any other relevant details.
Hope this helps,
-Bill