I think it's better to always use ApplicationContext, unless you're in a mobile environment like someone else said already. ApplicationContext has more functionality and you definitely want to use the PostProcessors such as RequiredAnnotationBeanPostProcessor, AutowiredAnnotationBeanPostProcessor and CommonAnnotationBeanPostProcessor, which will help you simplify your Spring configuration files, and you can use annotations such as @Required, @PostConstruct, @Resource, etc in your beans.
Even if you don't use all the stuff ApplicationContext offers, it's better to use it anyway, and then later if you decide to use some resource stuff such as messages or post processors, or the other schema to add transactional advices and such, you will already have an ApplicationContext and won't need to change any code.
If you're writing a standalone app, load the ApplicationContext in your main method, using a ClassPathXmlApplicationContext, and get the main bean and invoke its run() (or whatever method) to start your app. If you're writing a web app, use the ContextLoaderListener in web.xml so that it creates the ApplicationContext and you can later get it from the ServletContext, regardless of whether you're using JSP, JSF, JSTL, struts, Tapestry, etc.
Also, remember you can use multiple Spring configuration files and you can either create the ApplicationContext by listing all the files in the constructor (or listing them in the context-param for the ContextLoaderListener), or you can just load a main config file which has import statements. You can import a Spring configuration file into another Spring configuration file by using which is very useful when you programmatically create the ApplicationContext in the main method and load only one Spring config file.