Dart's Web Component vs Angular's Directive
They serve similar purposes indeed. However Web Components are not a part of Dart, they are a part of browsers. Dart Web-UI does use Web Components (http://www.w3.org/TR/2013/WD-components-intro-20130606/). AngularJS has plans to use them for 2.0 as can be read here: http://www.2ality.com/2013/05/web-components-angular-ember.html
What it comes down to is that a lot of the functionality is shared. But Angular.js has a pure JS implementation and Darts' Web-UI uses browser techniques.
Angular will use the underlying web platform features available to it (e.g. Node.bind, template integration, Custom Elements, etc...)