This is pretty clearly explained in the Spring documentation about the Container Extension Points.
The BeanPostProcessor interface defines callback methods that you can
implement to provide your own (or override the container’s default)
instantiation logic, dependency-resolution logic, and so forth. If you
want to implement some custom logic after the Spring container
finishes instantiating, configuring, and initializing a bean, you can
plug in one or more BeanPostProcessor implementations.
So in essence the method postProcessBeforeInitialization defined in the BeanPostProcessor gets called (as the name indicates) before the initialization of beans and likewise the postProcessAfterInitialization gets called after the initialization of the bean.
The difference to the @PostConstruct, InitializingBean and custom init method is that these are defined on the bean itself. Their ordering can be found in the Combining lifecycle mechanisms section of the spring documentation.
So basically the BeanPostProcessor can be used to do custom instantiation logic for several beans wheras the others are defined on a per bean basis.