They really accomplish similar goals. However, web components are not part of Dart; they are part of browsers. Dart Web-UI uses web components ( http://www.w3.org/TR/2013/WD-components-intro-20130606/ ). AngularJS plans to use them for 2.0, as you can read here: http://www.2ality.com/2013/05/web-components-angular-ember.html
What happens is that many functions are separated. But Angular.js has a pure JS implementation, and the Darts web interface uses browser methods.
Angular will use the underlying web platform features available to it (eg Node.bind, template integration, Custom Elements, etc...)