A Servlet and a Spring MVC Controller can be used to do the same thing but they act on different level of a Java Application
The servlet is a part of the J2EE framework and every Java application server (Tomcat, Jetty, etc) is built for running servlets. Servlet are the "low level" layer in the J2EE stack. You don't need a servlet.jar to run you application because it's prepackaged with the application server
A Spring MVC controller is a library built upon the servlet to make things easier. Spring MVC offers more built-in functionalities such as form parameter to controller method parameter mapping, easier handling of binary form submissions (i.e. when your form can upload files). You need to package the required jars to your application in order to run a Spring MVC Controller
You should use a servlet when you need to go "low level", and example could be for performance reason. Spring MVC performs good but if it has some overhead, if you need to squeeze out all you can from your application server (and you have already tuned the other layers such as the db) go with a servlet. You can choose a servlet if you want to understand the foundation of the J2EE web specifications (i.e. for educational purposes)
In all the other cases you can/should choose a web framework. Spring MVC is one of them; with Spring MVC you don't need to reinvent the wheel (i.e binary form management, form parameter to bean conversion, parameter validation and so on).
Another plus of Spring MVC is that in one class you can easily manage input from different urls and methods, doing the same in a servlet is possible but the code is more complicated and less readable.
My opinion is that Spring MVC is good for building rest services and managing simple applications (i.e web application with simple forms). If you need to manager very complex forms with Ajax, nested forms, and an application with both session and page state my advice is to switch to a component based framework (like apache wicket for example).