I believe that we have much more PROS than CONS. One of the good ideias to give HTML pages trought a WEB app server, like apache, nginx, or ISS, is that you can apply more security and control to the container that is delivered.
BUT, HOWEVER
Use static content, like JS, CSS, and HTML5 to consume only services, is the next goal in software development. When you start to divide things like, API and UX, you can test then separatly, you can develop at same time, service and interface, and you have much more speed and quality to development.
When we look at a web page, and the weight that the DOM have, and how much cost for the app server to give all that container to the user, and sometimes less then 10% of this is JSON from a service, we need to start to rethink the architecture of our web app.
I have been developing apps like this since one year and a half now, all the projects, and be sure, we re not to come back to the past. Its very important to work oriented to services, and how to consume this services.
If you use for example, Amazon S3 to host your HTML,JS, CSS, IMAGES, files, you dont need a app server, only the REST api to consume and give the content to the user. Costless and very,very faster.