What is the difference between aircraft integration and assembly?
I was many years with an OEM.  Integration is process of coordinating and incorporating, harmonizing you might say, the unique engineering of systems and processes of two engineering and production organizations or products, to make them work together in the final product.  To integrate them.
For example, a manufacturer designs and builds an airframe.  An avionics suite, or some other system, designed and manufactured by a second party, not specifically for a given type, is selected to be used in this new airframe.
Someone has to integrate, or adapt, the pre-existing avionics suite into the airframe and systems of the host.  The integrator is responsible for all of the activities of incorporating that system;  where/how it fits, how it interfaces with host systems, etc. In other words, the integrator is responsible for making the second party system work smoothly in the host system.