TL;DR
Both methods mentioned are not mutually exclusive. Either is fully capable of producing a web page for your users. If you need higher interactivity, you will have to add code to your page, using a framework or some combination with your own code. Your needs will determine which method or combination you require to build the web page.
No matter which of the methods used to create a web page you use, the rendering is done by the browser.
Server Page Generation
Some history
This may be old school (as of May 2014), but it is a tried and true method. The origins of this method started with Common Gateway Interface way back in 1993 (maybe earlier). Using CGI and perl scripts, a request for a web page could do something more than just feed a static HTML page to a browser. CGI has evolved into Java Server Pages, ASP.NET, PHP, and several other server based page generation technologies.
The Good
You can generate pretty much anything you want to send to the client. Pictures, documents, security encryption keys, etc. You can even generate HTML pages with dynamic content using Javascript on the page. There are some systems that don't store HTML pages anymore, but build the page on the fly (DotNetNuke is one example). It is possible to build a Single Page Application (SPA) this way.
The Bad
It is easy to fall into the trap that you have to make round trips to the sever to process every change on a page. You can dramatically increase the load on your server by poor implementation, although this is true of both methods.
The Ugly
Trying to mix Server Page Generation and SPAs can be a nightmare to build and debug. You have some code that is built on the server, but executes on the browser. A little planning will go a long way to mitigate this issue. Code reviews are also a good idea. If your junior developer can't follow what you are doing, your code might be overly complex.
RESTful Interface To A Server
Some (more) history
This is a relatively new technique (less than 5 years old). It is one method to make a web page more "reactive" to user interaction. Using AJAX as a transport mechanism between the browser and the server, along with some code on the web page to process the data, the user gets a web page that acts more like a desktop application.
The Good
This method of page generation allows for frameworks to be used in the construction of your interactive web page. The frameworks have advantages of being open source, heavily developed, heavily tested, and (maybe) documented. This method is not mutually exclusive with Server Page Generation. You will find that an application may appear to be fully SPA, but isn't (e.g., Google's GMail).
The Bad
Javascript not functioning on a browser. All of the browser variations that you have to support. You should have a static HTML page as a fallback for when the browser doesn't support your fancy framework (think text based browser, especially for the blind).
The Ugly
This is the same as for Server Page Generation.