As I understand it, "physical" and "functional" are just two different ways of analyzing an interaction between proteins.
Proteins can and do interact physically, meaning they can bind in specific ways and sites, and this binding can produce changes in those same proteins (like conformational changes), which alter their properties. This is the purely physical analysis of the protein interaction. The functional analysis begins when we ask the questions "What is the role that this interaction performs in the cell/organism/living system?"; "What effects do these changes in the protein are going to create in the cell/organism/living system?". Often a conformational change in a protein induced by an interaction with another activates or inactivates it. In other words, an interaction can create a conformational change that enables (activation) or disables (inactivation) the protein to catalyze a given reaction, and so the referred interaction plays an important function in regulating the protein's activity. For instance, phagocytes (cells capable of phagocytosis) have in their membranes proteins capable of physically interacting with proteins present in pathogens membranes. They physically bind to them, changing their conformation in the process. This is the physical part of the interaction, it is what happens purely physically (and can be studied more deeply). This conformational change will enable the protein to catalyze a series of reactions inside the phagocytic cell that will ultimately lead to the phagocytosis and destruction of the pathogen. This is the functional analysis of the interaction, it is understanding what is functionally achieved by the interaction, in this case enabling phagocytosis (and can be studied more deeply).
