Nancy, like Sinatra, is a general-purpose web framework. You could implement an API in it, or you could implement a user-facing website.

Nancy and Sinatra are typically used for smaller sites. They're based around routes, controllers, and views but they don't (for better or worse) lock you into an MVC project structure like ASP.Net MVC or Rails.

Web API, by comparison, is pretty much "just for implementing a web-based API" and not something you'd want to use for a user-facing web application that serves up HTML pages.

ServiceStack [1] is a closer comparison to Web API. A lot easier to set up and runs in Mono, but not free (although it is open source [2]).

[1] - http://servicestack.net [2] - https://github.com/ServiceStack/ServiceStack