Website structure is how your website is designed as far as the scripting and also how easy and efficiently you can navigate your website using the menu items and internal links. The ultimate focus here should be on creating the best user experience (UX) possible while balancing and abiding by the latest in quality design and coding principles.
Layout involves how you organize your content on each page, which communicates to the search engines what is the most important material and what should users look at first and foremost. Think of it as a teacher instructing the students what material and concepts are the most important to grasp for the exam. By implementing SEO tags and micro data, you help search engines better understand the context of your content.
Sometimes the search engine crawlers (also called robots, spiders, bots etc.) need a bit more information in order to index (record and interpret for search engine results) correctly. This is where robot commands, schema, 301 redirects, sitemaps, image maps, keywords, descriptions, and plenty of other useful SEO tools come in handy.