Using asynchronous call can free up your application to do other things while waiting for the response. Since there is a fairly large amount of time (in computer cycles) waiting for a web server to respond, that time can be used for better things such as displaying a status update or doing some other work.
For example, if you had a program that performed a complicated calculation and a step of that calculation included using some reference data from a remote web service. By calling the web service asynchronously at the start of the calculation, continuing the parts of computation that can be performed locally, and then using the result of the web service call when it is available to complete the computation you can reduce the overall time of the calculation.
Since your application code is not blocked waiting for the web service to respond, you are able to utilize that wait time to the benefit of the user.
Another reason is scaling, particularly in web sites that make calls to other web services. By using asynchronous page methods (or tasks), IIS can scale your application more effectively by deferring your pages that are waiting on asynchronous web requests to whats known as an "IO thread", freeing up the main ASP.NET worker threads to serve more web pages.