Wikipedia teaches us that problems in Ramsey theory typically ask a question of the form: "how many elements of some structure must there be to guarantee that a particular property will hold?"
It also teaches us that "Extremal graph theory studies extremal (maximal or minimal) graphs which satisfy a certain property". This sounds very similar. Let's take Turan's theorem (the basic example of an EGT theorem) - it gives a bound on the number of edges in a clique-free graph (for a given clique size). Put differently, it tells us how many elements (edges) of a certain structure (graph with fixed number of vertices) must be there to guarantee that a particular property (existence of a clique of a specific size) will hold.
One can argue that in EGT we try to understand the structure of the extremal objects, but I don't see why Ramsey theory should not try to do the same thing, and also I don't see how this is done in the (beautiful) Erdős–Stone theorem which is considered a fundamental theorem of EGT.
So is there a real difference or is it just a matter of taste?