web.config is used with web applications. web.config by default has several configurations required for the web application. You can have a web.config for each folder under your web application.
app.config is used for Windows applications. When you build the application in Visual Studio, it will be automatically renamed to <appname>.exe.config and this file has to be delivered along with your application.
You can use the same method to call the app settings values from both configuration files: System.Configuration.ConfigurationSettings.AppSettings["Key"]