An applet runs in an access-limited sandbox within a browser. Examples include specialized calculators, popup info displays, data gathering interfaces, etc.
A servlet resides behind a web server and has no security restrictions. They are frequently "faceless"; there is no visual representation of the servlet on the web page. Whether faceless or not, servlets usually return HTML to the web server for return to the client browser. Examples of servlets include mass data manipulators (database-intensive queries), AJAX handlers, web services, etc.