I think it's not about ASP.NET MVC with Angular. It should be Angular or without Angular.js. Still it would make more sense to use Angular.js with ASP.NET Web API as Angularjs can handle UI part efficiently.
Here are few advantages and disadvantages of Angualr.js
Advantages:
- Clean way of creating UI in JavaScript
- Angular.js is testable. You can write test against it.
- Angular js provides reusable components
- With angularjs develoeprs can be more productive.
Disadvantages:
- Not secure. As most of the code related UI will be in Javascript only
- Not SEO friendly as most of the UI code in javascript only
So it's dependents on nature of application. You need to create application that more secure or you need to create a application like E Commerce site where SEO is more important. Then you should go with plain ASP.NET MVC. But if that is not the option then it's worth using angular.js.
You can also have hybrid types of application where if you require secure pages or SEO related then you can use ASP.NET MVC and for others like user profile you can use angular.js
So it's tradeoff you need to make based on your requirement. Both have pros and cons.