I would argue that there are only two important questions when considering which of the two approaches to take: what are you building, and who you are building it for. Everything else falls behind. Points like "but than I will have to build two applications instead of one"... Common, It's just a matter of perspective. It doesn't really matter if you call it two applications or two modules or two battery staples, as long as it does what you imagined in the best possible way for the user.

If you are building an interactive web application that should be akin to classic desktop applications, you should really build a separate UI, just like if you were building a desktop application. If you are using a framework such as GWT, than it doesn't even have to "look" like two different applications from the developers point of view. Building a detached UI doesn't imply creating a nice public API at the same time.

On the other hand, if you are trying to create a content based site logically designed as linked pages of content, there really is no good reason to brake the classic web page layout and reinvent the wheel.

The problem, ofc, is what to do when your site is a hybrid between the two. Well, maybe a hybrid approach should apply also?

Most of the problems mentioned with the newer approach, js UI, are technical problems that must be solved (by us), not conceptual problems in the approach itself.