You should not be comparing Knockout.js to Asp.NET MVC. The only similarities are that they both use the same Model-View-Controller pattern.
Knockout.js:
Knockout is a standalone JavaScript implementation of the Model-View-ViewModel pattern with templates. The underlying principles are therefore:
a clear separation between domain data, view components and data to be displayed
the presence of a clearly defined layer of specialized code to manage the relationships between the view components
The latter leverages the native event management features of the JavaScript language.
Asp.NET MVC:
ASP.NET MVC allows you to build a web application as a composition of three roles: Model, View and Controller.
A model represents the state of a particular aspect of the application. A controller handles interactions and updates the model to reflect a change in state of the application, and then passes information to the view. A view accepts necessary information from the controller and renders a user interface to display that information.
As you can see Knockout.js is an MVC implementation for JavaScript while Asp.Net MVC is an MVC implementation for a complete web application from front end to server. These technologies can even (and very often) be used together to build well structured applications on both the front end and the back end.