Dashboards provide a flexible layout for your data exploration - something similar to what you are used to in traditional BI tools. You will have a multi-page canvas (landscape or vertical A4) where you have full control over the layout. You will be able to cross-filter the entire canvas accross any widgets which share a common data source. Take a look at an example dashboard here.
Data stories allow you to go into a specific story line within the data without the reader having to drill the data on their own. In a visually attractive and intuitive way you will be able to explain a topic through graphs, maps, charts and text and the reader will have it presented in the form of a story or article. Take a look at a example data story here. You can share the created content by embedding, sharing a full story as a URL link or we will convert it into a video for you.
Within a Data Story you as the creator are in control of the filters of your widgets. As the reader scrolls through the story they get the highlights you (the creator) consider essential / interesting.