I got two websites, both use Wordpress, same plugins, same server but for some reason Google lists them in two different ways. I am wondering why are there extra links from the website underneath the main url (blue boxes) such as here:
But it doesn't show up on this website:
The reason will be simply that the ranking algorithm has not detected that showing them would be useful to the searcher based on the search query being used.
Links which are shown below the search results is sitelinks, and which are primarily meant to help visitors or searchers navigate your site quite easily. Google analyzes your website link structure and find shortcuts that will save users time and allow them to quickly find what they're looking for. This sitelinks is automatically generated by google and you don't have the provision to change it. Only thing you can do is if you don't want any of the site links to be listed then you can demote it in the google webmaster tools.
This topic is discussed in the FAQ: GoogleSticky thread.