They are very different concepts also if their names are similar.
A Servlet Container or Web Container (like Tomcat) is an implementation of various Java EE specifications like Java Servlet, JSP, etc. Put in a simple way, it is an environment where Java web applications can live. A web server + Java support.
A Spring Container on the other hand, is the core and the engine of the Spring Framework. It is an IoC Container that handles Spring applications lifecycle creating new beans and injecting dependencies.
Because a Spring application can be a web application, a Spring Container can "live" inside a Web Container.
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