Servlet object has a well-defined life cycle where creation and initialization steps are distinct. Usually you don't want to do heavy and unreliable work in constructors, e.g. getting DB connection pool or initializing cache. You have a chance to do it after a Servlet object is successfully created.
Also, here http://oreilly.com/catalog/jservlet/chapter/ch03.html is a historical reason:
The init() method is typically used to perform servlet initialization--creating or
loading objects that are used by the servlet in the handling of its requests. Why not use a constructor instead? Well, in JDK 1.0 (for which servlets were originally written), constructors for dynamically loaded Java classes (such as servlets) couldn't accept arguments.
So, in order to provide a new servlet any information about itself and its environment, a server had to call a servlet's init() method and pass along an object that implements the ServletConfig interface.
Also, Java doesn't allow interfaces to declare constructors. This means that the javax.servlet.Servlet interface cannot declare a constructor that accepts a ServletConfig parameter. It has to declare another method, like init(). It's still possible, of course, for you to define constructors for your servlets, but in the constructor you don't have access to the ServletConfig object or the ability to throw a ServletException.