In brief, web server handles on request and responses of a http protocol. Where as an application server, provides many other things apart from request and responses. It provides services like transaction management.
All application servers are web servers and all web servers need not be application servers. :))
Web Server is a computer that serves web pages. Web server has its own IP adress and also a domain name.
Application server is a computer that handles all the applications or backend applications of an organisation between the users. Applications servers are mostly used for transaction based complex applications.
A Web server handles HTTP requests, whereas an application server handles business logic to application programs through any number of protocols. I think this the main difference between them.