On page seo is focused on your actual website from page load speed right through to whether what you have on your site is unique or not.
In google the algorithm that checks your on page is google. Bad on page seo makes search engine bot think that your site is spam whereas having a good website with good on page seo makes your site seem less likely to be spam.
Site design is something the panda update checks. Does the site have a responsive design? Having a design which is responsive means that not just desktop users can enjoy your site but also mobile users too.