Paul Stovell discusses the transition of Octopus to an API-first approach, opting to rebuild the entire UI using Angular.js, Twitter Bootstrap, Angular UI-Bootstrap, Underscore, and jQuery, which are better suited for the newly adopted API. The development process involves using WebStorm for coding and Grunt, which runs under Node, for building the application by compiling HTML templates, concatenating and minifying JavaScript files, and combining CSS files into a single output. Stovell highlights the advantages of this approach, including an improved development experience and the necessity of using the public API like any other user. He shares progress on the UI for Octopus 2.0, emphasizing improvements such as a responsive design, the ability to reuse NuGet packages from previous releases, and enhanced filtering options when searching for old packages.