The main difference is that Lightning Apps are outside the main Lightning Experience (LEX) that is Salesforce. Think of them as standalone Visualforce pages. They are used to create a custom Lightning experience that doesn't have the LEX UI.
Lightning App Builder (LAB) is a graphical interface that allows you to compose apps using any number of standard and custom components without writing code. LAB allows you to override the default behavior of tabs, record detail pages, and even create new tabs that are driven by standard and custom components. These will always be in the LEX UI.
In general, you can't use Lightning Apps in anything. They're not meant to be used that way. If you need something that's reusable across different experiences, it must be a component. Components are the core structure of the Lightning Experience. Also, there's a limit of one "app" per page (the Lightning App variety), which is why you can't put apps in tabs or generally use them anywhere in the LEX UI, because they would conflict with the "one.app" (so-called because LEX's precursor was Salesforce 1).